Primitive Permutation Groups with Soluble Stabilizers and Applications to Graphs

Hua Zhang

School of Mathematics and Statistics

A thesis submitted for the degree of Doctor of Philosophy of the University of Western Australia

February 2011 September 13, 2011 Dedicated to my parents

ii Statement

The results in this thesis are my own except where otherwise stated.

iii Acknowledgements

First and foremost, I would like to express my sincere gratitude to my supervisor, Professor Cai Heng Li, both for his constant and patient assistance and advice throughout the preparation of the thesis, for his continuous encouragement, and for all his kindly help and emotional support during my stay here in Australia. I would like to express my sincere gratitude to my supervisor, Winthrop Pro- fessor Cheryl E. Praeger. As one of the most influential mathematician in the world, she has led the research direction of many of her Ph.D students, and each of them benefits from her rich experience and vast knowledge. Her friendliness and long-term support will be most appreciated. It is a tremendous fortune to do my Ph.D under the supervision of the two leading mathematicians. I gratefully acknowledge the financial support of the Scholarships for Interna- tional Research Fees (SIRF) from the University of Western Australia. I am deeply indebted to my wife Yan Cao and my son Yucheng Zhang for their understanding, unlimited love and support. I am very grateful to Dr Jian Ping Wu, for her long-term friendship, so much help and spiritual support. I would like to thank Associated Professor Michael Giudici and Associated Pro- fessor John Bamberg for their friendship, constructive suggestions for the thesis, and the assistance of computation.

iv Abstract

In the past 60 years interactions between group theory and the theory of graphs have greatly stimulated the development of each other, especially the theory of symmetric graphs (or more generally vertex-transitive graphs) has almost developed in parallel with the theory of permutation groups. The aim of this thesis is to make an effort, both in terms of pure research and broader value, to solve some challenging problems in these two fields. First we considered the problem of classifying finite primitive permutation groups with soluble stabilizers. This problem has a very long history. By the O’Nan- Scott Theorem, such groups are of type affine, almost simple, and product action. We reduced the product action type to the almost simple type, and for the latter, a complete classification was given, which forms the main result of the thesis. The main result was then used to solve some problems in algebraic graph theory. The first application of which is to classify edge-primitive s-arc-transitive graphs for s ≥ 4. After undertaking a general study on the local structures of 2-path-transitive graphs, we presented the second application by classifying finite vertex-primitive and vertex-biprimitive 2-path-transitive graphs. The result then helps us to be able to construct some new half-transitive graphs. Another application of the main result is that a complete classification of finite vertex- biprimitive edge-transitive tetravalent graphs is given (recall that for the cubic case, the classification was given by Ivanov and Iofinova in a highly cited article in 1985).

v Publications arising from this thesis

1, C. H. Li and H. Zhang, The ﬁnite primitive groups with soluble stabilizers, and s-arc-transitive graphs with s ≥ 4, Proc. London. Math. Soc. (3)103 (2011), 441-472. (based on Chapter 4 and Chapter 5). 2, C. H. Li and H. Zhang, On 2-path-transitive graphs, submitted to J. Graph Theory (based on Chapter 6). 3, C. H. Li and H. Zhang, Finite vertex-primitive and vertex-biprimitive 2-path-transitive graphs, submitted to J. Algebraic Combin. (based on Chapter 7). 4, C. H. Li and H. Zhang, Finite vertex-biprimitive edge-transitive tetravalent graphs, submitted to J. Combin. Theory, Ser. B (based on Chapter 8).

vi Contents

1 Introduction1 1.1 Introduction...... 1 1.2 Literature review...... 3 1.3 Main results and the structure of this thesis...... 8

2 Preliminaries 10 2.1 Permutation groups...... 10 2.2 Orbits, stabilizers, and transitivity...... 12 2.3 Blocks, primitivity, and the O’Nan-Scott Theorem...... 14 2.3.1 The O’Nan-Scott Theorem...... 15 2.4 More on almost simple groups...... 18 2.5 The subgroup structure of the classical groups...... 20

3 Graphs 22 3.1 Basic concepts of graphs...... 22 3.2 Symmetries of graphs...... 23 3.3 General constructions of graphs...... 26

4 Finite primitive groups with soluble stabilizers. 31 4.1 The main result...... 31 4.2 A reduction for the proof of Theorem 4.1.1...... 33 4.3 Subgroups of classical groups of Lie type...... 35 4.3.1 Preliminaries...... 36

4.3.2 Soluble maximal Ci-subgroups...... 37 4.3.3 The maximality...... 43 4.4 Subgroups of linear groups...... 45 4.5 Subgroups of symplectic groups...... 51 4.6 Subgroups of unitary groups...... 53 4.7 Subgroups of orthogonal groups...... 56 4.8 Tables...... 58

1 Contents

5 Finite edge-primitive s-arc-transitive graphs with s ≥ 4 66 5.1 The classiﬁcation...... 66 5.2 A reduction for the proof of Theorem 5.1.1...... 67 5.2.1 s-Arc transitive graphs...... 67 5.2.2 Edge-primitive graphs...... 69 5.3 Proof of Theorem 5.1.1...... 70

6 Finite 2-path-transitive graphs 74 6.1 The problem and the results...... 74 6.2 Proofs of Theorems 6.1.1 and 6.1.2...... 76 6.3 Line graphs...... 80 6.4 Proof of Theorem 6.1.4...... 81

7 Finite vertex-primitive and vertex-biprimitive 2-path-transitive graph- s. 86 7.1 The main results...... 86 7.2 A reduction...... 88 7.3 Examples...... 90 7.4 Proof of the main theorems...... 96 7.4.1 Proof of Theorem 7.1.1...... 96 7.4.2 Proof of Theorem 7.1.2...... 98 7.4.3 Proof of Theorem 7.1.3...... 100 7.5 The numbers of the non-isomorphic graphs...... 100

8 Finite vertex-biprimitive edge-transitive tetravalent graphs 104 8.1 The classiﬁcation...... 104 8.2 Examples...... 106 8.3 Local structure of edge-transitive tetravalent graphs...... 109 8.4 A reduction...... 111 8.5 Almost simple type...... 114 8.6 Proof of Theorem 8.1.1...... 120

9 Bibliography 124

2 Contents

Nomenclature

Groups

Fq Field with q elements G, H, K Groups n Arbitary soluble group of order n Zn Cyclic group of order n G0 Derived group of G G.H Extension of G by H G:H Split extension of G by H |G:H| Set of right cosets of H in G NG(H) Normalizer of H in G CG(H) Centralizer of H in G Out(G) Outer automorphism group of G Sn Symmetric group of degree n An Alternating group of degree n Gα Stabilizer of α in G G(∆) Pointwise stabilizer of ∆ in G G∆ Setwise stabilizer of ∆ in G AGLn(q) Aﬃne groups of dimension n over Fq PSLn(q), PSUn(q) Classical groups of Lie type of dimension n over Fq

3 Contents

Graphs

Γ , Σ Graphs V Γ Vertex set of Γ EΓ Edge set of Γ AΓ Arc set of Γ AutΓ Full automorphism group of Γ V al(Γ ) Valency of a regular graph Γ Cn Cycle of degree n Kn Complete graph of degree n Kn,m Complete bipartite graph of degree n + m Γ (α) Neighborhood of α [1] Gα Kernel of Gα acting on Γ (α) Γ(α) Gα Induced permutation group of Gα on Γ (α) ΓN Quotient graph of Γ with respect to normal subgroup N Cos(G, H, HgH) Coset graph Cos(G, L, R) Double coset graph

4 Chapter 1

Introduction

1.1 Introduction

In the past 60 years interactions between group theory and the theory of graphs have greatly stimulated the development of each other, especially the theory of symmetric graphs (or more generally vertex-transitive graphs) has almost developed in parallel with the theory of permutation groups. The study of symmetric graphs has long been one of the main themes in algebraic graph theory. In early work in [95, 99, 103, 119, 120, 128, 133], finite group theory was used in the study of vertex-transitive graphs and many surprising results were obtained; while in [95, 106, 124, 129] ideas derived from graph theory were used to obtain strong and deep results about permutation groups, and new descriptions of some finite simple groups, and more importantly some new and interesting classes of permutation groups were found. Following the Classification of Finite Simple Groups (henceforth referred to as CFSG), together with the O’Nan-Scott Theorem about the finite primitive groups, more and more classes of symmetric graphs have been completely determined. In recent years symmetric graphs with additional properties, such as vertex-primitive or quasiprimitive graphs, locally primitive graphs, and s-arc-transitive graphs for s ≥ 2, just to name a few, have been extensively studied, see [41, 43, 44, 68, 69, 95, 97]. However, many challenging problems in this field still remain unsettled. The aim of this thesis is to make an effort towards this direction. The study will mainly be centered around the following aspects:

• adding new information in permutation group theory, and

• extending the knowledge of symmetric graphs.

By deﬁnition a graph Γ is called symmetric if its automorphism group acts transitively on the set of the arcs. For some positive integer s, Γ is called s-arc-transitive if the automorphism group acts transitively on the set of s-arcs. 1-arc-transitive is just

1 1.1. Introduction symmetric. The study of s-arc-transitive graphs has a long and fruitful history, going back to the classical work of Tutte in 1947, and is currently an active topic in algebraic graph theory. After Tutte (1959, [120]) proved his seminal result that there exist no ﬁnite cubic s-arc-transitive graphs for s ≥ 6, Weiss (1981, [128]) generalized Tutte’s result by proving that there exist no ﬁnite s-arc-transitive graphs of valency greater than two for s ≥ 8. Since then, s-arc-transitive graphs have received extensive attention and one of the central issues in algebraic graph theory is this problem: constructing and characterizing s-arc-transitive graphs for s ≥ 2, in particular, for s ≥ 4. Regarding this problem, considerable work has been done and some results are available in the literature (see the next section for the review). It is a fact that up to now, the known families of s-arc-transitive graphs, for s ≥ 4, are rare, and the task of seeking new families of s-arc-transitive graphs has never ceased. The initial motivation of this thesis is to solve the following problem proposed by Weiss in 1999:

Problem 1: Classify ﬁnite edge-primitive s-arc-transitive graphs for s ≥ 4.

Here a graph Γ is called edge-primitive if its automorphism group AutΓ acts primitively on the set of the edges. For edge-primitive s-arc-transitive graphs, where s ≥ 4, it is known that the stabilizers of their edges are soluble (see [38, 126]). Therefore to solve problem 1, we are naturally lead to studying another important problem:

Problem 2: Determine all ﬁnite primitive permutation groups with soluble point stabilizers.

This latter problem also has a long and rich history, and is very interesting in its own right, as shown in the literature review given in the next section. In this thesis we obtain a satisfactory result about this problem, which turns out to be a powerful tool in solving certain problems related to graphs. In fact the outcome of studying Problem 2 can not only be used to ﬁnd an answer for Problem 1, but also produces several important applications in symmetric graphs. The second application is on the study of an interesting class of locally primitive graphs, namely, the 2-path-transitive graphs. Let Γ be a connected vertex-transitive graph with G being an automorphism group of Γ. We say that Γ is (G, 2)-path transitive if G is transitive on the set of 2-paths of Γ. Suppose that Γ is (G, 2)-path transitive but not (G, 2)-arc transitive. Then for a vertex

α of Γ, the point stabilizer Gα is 2-homogeneous but not 2-transitive on Γ(α), the set of vertices in Γ which are adjacent to α. It follows that Gα has odd order (see [24, page

35]) and so is soluble. Suppose further that Gα is maximal. Then the pairs (G, Gα) is explicitly known by the study of Problem 2. Consequently the solution of Problem 2 enables us to completely solve the following problem in this thesis:

2 1.2. Literature review

Problem 3: Classify ﬁnite vertex-primitive and vertex-biprimitive 2-path-transitive graphs which are not 2-arc-transitive.

A transitive permutation group G is said to be quasiprimitive if every nontrivial normal subgroup of G is transitive. In 1993 Praeger generalized the O’Nan-Scott Theorem for finite primitive groups to quasiprimitive groups and proved that finite quasiprimitive groups have the same types as that of finite primitive groups. A group G is said to be biquasiprimitive if it is not quasiprimitive and each of its normal subgroups has at most two orbits. Similar to the argument in [95], the study of 2-path-transitive graphs can be reduced to the study of vertex-quasiprimitive and vertex-biquasiprimitive graphs. Thus it is possible to extend Problem 3 to vertex-quasiprimitive and vertex-biquasiprimitive cases. For this reason we carried a study of the vertex-quasiprimitive case. One of the main motivations of studying 2-path-transitive graphs is that there is a natural connection between 2-path-transitive graphs and half-transitive graphs (recall that a graph is called half-transitive if it is vertex- and edge-transitive, but not arc- transitive). Let Γ be a graph with vertex set V Γ and edge set EΓ. The line graph of Γ is defined to be the graph with vertex set EΓ, such that two vertices are adjacent if and only if they are incident in Γ. It can be shown (see chapter 5) that a graph Γ is (G, 2)-path-transitive but not (G, 2)-arc-transitive if and only if its line graph is G-half-transitive. Thus a solution to Problem 3 leads a new approach of constructing half-transitive graphs, and the constructions comprise an important part of this thesis. In graph theory, graphs with small valencies have received special attention in the literature. When studying these small valency graphs, usually extra restrictions should be added. One way of adding such restrictions is by assuming the vertex-primitivity or the vertex-biprimitivity. Finite vertex-biprimitive edge-transitive cubic graphs were classified by Ivanov and Iofinova in 1985. As the last, and certainly not the least, application of the outcome of Problem 2, in this thesis we completely solve the following problem:

Problem 4: Classify ﬁnite vertex-biprimitive edge-transitive tetravalent graphs.

1.2 Literature review

Centering around the problems proposed in the last section, the literature review given in this section will be focused on the developments of ﬁnite primitive groups with soluble stabilizers; some early work on symmetric graphs, in particular, on highly symmetric graphs; recently developments of half-transitive graphs; and a few known results on 2-path-transitive graphs and biprimitive graphs of small valency.

3 1.2. Literature review

Permutation groups arose from the study of roots of polynomials, but soon became objects of independent interests. The earlist study of finite primitive groups dates back to Ruffini’s work in 1799. Also worthy of special mention is Jordan’s work in 1870, which described primitive soluble subgroups of the linear groups of the form GLrt (Fq), with r prime. Following Jordan was a lot of early work in the last years of the 19th century and the first years of the 20th century. Early researchers were mainly interested in listing all permutation groups (including primitive permutation groups) of a small given degree. Attention was also paid to finite primitive groups with soluble stabilizers. Reitz in 1904 studied the finite primitive groups with odd order [100]. It is known that if G is a finite primitive group with a suborbit of small length (no larger than 4), then the point stabilizer Gα is always soluble. It can be read off from Wielandt’s book [131] that if G is a finite primitive group with a suborbit of length 2, then G is a dihedral group of order 2q (q prime). The 2-transitive soluble groups were determined by Huppert [51] in 1957. In [133] Wong determined all primitive groups with a suborbit of length 3, and in [99, 107, 121] primitive groups with a suborbit of length 4 were determined. In [65] Knapp studied the structure of the point stabilizers of primitive groups for a large class of subconstituents. In [81] Liebeck and Saxl classified primitive permutation groups in which a point stabilizer has odd order. Primitive permutation groups with soluble stabilizers also naturally appear in other mathematical problems, in particular, in algebraic graph theory. For instance, if a map on a 2-manifold is vertex-primitive, then the automorphism group of the map is primitive and has a dihedral vertex stabilizer. In order to obtain a classification of vertex-transitive embeddings of complete graphs, a classification of primitive permutation groups with dihedral point stabilizers was needed and given in [75], and generalized a result of Wang [122], where primitive groups with dihedral point stabilizers of order 2p (p prime) were determined. In[123] Wang determined all primitive groups G with a sharply 2-transitive subconstituent, which includes the case that G is almost simple and the point stabiliser

Gα is soluble. Another example is that the vertex-stabilizers of 3-arc-regular graphs are soluble (see [76] for a recent classiﬁcation of vertex-primitive and vertex-biprimitive 3-arc-regular graphs). It follows from the O’Nan-Scott Theorem that a primitive group with soluble stabilizers is of type aﬃne, almost simple, and product action (see chapter 2 for details). If

G is a primitive group of type aﬃne, then Gα is solvable if and only if G is soluble. Jor- dan’s work in 1871 essentially gives a table containing the numbers of conjugacy classes n 6 of maximal irreducible soluble subgroups of the linear groups GLn(p) for p < 10 . Then in the 40s and 50s of the last century, a lot of pioneering work had been done

4 1.2. Literature review

on the classification of irreducible soluble maximal subgroups of linear groups, mainly by Suprunenko and some other mathematicians from the former Soviet Union (see Suprunenko’s books [110, 111]). Building on Suprunenko’s work (and with due credit given to Jordan), in 1992 Short [105] listed almost all the soluble primitive groups of degree less than 256 (the list is incomplete as Alexander Hulpke detected that two conjugacy classes of monomial subgroups are missing). Short also discussed a general way of constructing the primitive affine groups. In [101] Roney-Dougal and Unger classified all affine primitive groups of degree less than 1000, and in [102] Roney-Dougal classified all primitive groups of degree less than 2500. In [29] Eick and Höflingpresented an algorithm which enables them to determine the solvable primitive permutation groups of degree at most 6561, and this list has been extended to degree 10000 by Höfling.In [34] Flannery and O’Brien published a complete and irredundant list together with an

electronic list of soluble irreducible subgroups of the linear groups GLn(F), where n is 2 or 3 and F is a finite field of characteristic greater than n. It seems intractable to list all the finite soluble primitive groups of type affine. For almost simple type, since the stabilizers of a primitive group form a unique conjugacy class of maximal subgroups, the classification of almost simple primitive groups with soluble stabilizers is closely related to a study of the (soluble) maximal subgroups of almost simple groups. During the last century, and even before, a substantial amount of work was devoted to finding the maximal subgroups of almost simple groups. The contents are so rich that a long and separate survey is needed to present the full sto- ry. Instead of giving such a review here, we will display most of the references on the maximal subgroups of almost simple groups in Chapter 4. We also refer to two survey papers [63, 77], both serve as excellent resources on the topic. The study of the primitive groups of product action type can naturally be reduced to the almost simple type, as can be seen in Chapter 4. As mentioned before, the study of s-arc-transitive graphs stems from Tutte’s work on cubic graphs in 1947. Tutte’s fundamental result greatly stimulated an extensive study on symmetric graphs and highly arc-transitive graphs. In [106, 107] Sims stressed the importance of the group theoretic method in graph theory, and generalized Tutte’s result considerably. In [45] Goldschmidt studied the point stabilizers and edge stabilizers of cubic edge-transitive (not vertex-transitive) graphs, he obtained an important extension of Tutte’s result, and introduced the so called “amalgam method”, which inspired a lot of work on groups and geometries. In [38] Gardiner studied s-arc-transitive graphs of valency p + 1, with p prime, and in [39] he studied s-arc-transitive graphs with doubly primitive point stabilizers.

5 1.2. Literature review

Weiss’s work [128] in 1981 is a landmark of the study of s-arc-transitive graphs, which shows that, apart from the cycles, there exist no finite s-arc-transitive graphs for s ≥ 8. Another paper [124] of Weiss serves as a good reference for the early work on s-arc-transitive graphs. One of the most important results in this study is the detailed description of the point stabilizers and edge stabilizers of s-arc-transitive graphs for s ≥ 2. This deep result, mainly due to Weiss and Trofimov, can be seen in the collected papers [114–117, 125–127]. In a series of papers, Biggs (partly with Hoare) discussed and constructed some 4-arc-transitive and 5-arc-transitive graphs (see [6–8]). In [13–16] Conder (partly with Walker) studied some s-arc-transitive graphs for s ≥ 4, including the constructions of an infinite family of 4-arc-transitive cubic graphs, an infinite family of 5-arc-transitive cubic graphs, and an infinite family of 7-arc-transitive graphs. In [54] Ivanov and Praeger classified vertex-primitive and vertex-biprimitive affine 2-arc- transitive graphs. In recent years, a wide range of study on s-arc-transitive graphs was undertaken by Li. In [69] he classified s-arc-transitive graphs of prime-power order, in [71] he studied s-arc-transitive graphs of odd order, and in [68] the finite vertex-primitive and vertex-biprimitive s-transitive graphs for s ≥ 4 were classified, together with a few newly constructed highly arc-transitive graphs. Another significant contribution of this study is made by Praeger on the structure of quasiprimitive groups. The importance of Praeger’s work relies on the fact that by adopting the normal quotient, the study of s-arc-transitive graphs can be reduced to the study of vertex-quasiprimitive s-arc-transitive graphs or bipartite graphs (see [95, 96]). Praeger’s work provides a general approach not only to the investigation of symmetric graphs, but also produces a natural impact on the study of other (large) families of graphs, such as locally-primitive graphs and locally-s-arc-transitive graphs. In the past twenty years the effectiveness of this approach is demonstrated by the fruitful results in this field. As the list is too long, we do not intend to present it here, and we refer to [97, 98] for references. The concept of half-transitivity was introduced by Alspach and Maruˇsiˇcin the latter part of the last century. The first examples of half-transitive graphs were found in 1970 by Bouwer [9], who constructed an infinite family of such graphs. However, the study of half-transitive graphs may date back to Frucht’s work [37] on the construction of a cubic 1-regular graph in 1952. In 1981 Holt [49] found a half-transitive graph with 27 vertices and valency 4. Then Alspach, Maruˇsiˇcand Nowitz [2] showed that Holt’s graph is the smallest one in the sense that there are no half-transitive graphs with fewer than 27 vertices or with degree less than 4. In the past twenty years the research on half- transitive graphs is an active area and a lot of results can be found in the literature. In particular, the survey paper [88] of Maruˇsiˇcis an excellent reference, where all the main

6 1.2. Literature review

results produced before 1997 were listed, along with a number of open problems. In the same paper he pointed out that the cases of vertex-primitive, the cases of small valency, such as valency 4 in particular, and the problems of classification, should receive special attention. Another paper which also serves as a nice reference is [66]. Some very recent work in this field can be seen in [67, 70, 72, 84, 108]. Interest on edge-primitive graphs lies on the fact that almost all of these graphs are symmetric. Furthermore, many famous graphs, such as the Heawood graph, the Tutte-Coxeter graph and the Higman-Sims graph, are edge-primitive. Unlike the vertex- primitive graphs, the existence of edge-primitive graphs is far more restrictive. Many edge-primitive graphs are implicitly listed in the ATLAS [19], these graphs admit a sporadic simple group as a group of automorphisms. In [130] Weiss determined all edge-primitive cubic graphs. In [44] Giudici and Li initiated a systematic study on edge-primitive and edge-quasiprimitive graphs. In terms of group theoretical language, they gave a necessary and sufficient condition for the existence of edge-primitive graphs, and then by using the O’Nan-Scott Theorem, they explored the possible edge and vertex actions of such graphs. For edge-primitive graphs, these are the only known references in the literature. The set of all s-path-transitive graphs forms a subclass of symmetric graphs, this subclass, in turn, includes the class of s-arc-transitive graphs as a subclass. In some sense, the concept of s-path-transitive graphs which are s-arc-intransitive is a generalization of half-transitivity. A basic study of s-path-transitive graphs which are not s-arc-transitive was undertaken in [17], where some basic properties on s-path-transitive graphs, in particular that of 2-path-transitive but not 2-arc-transitive graphs, are given. Up to date, this forms the only known reference to the author. A regular graph is called semi-symmetric if it is edge-transitive but not vertex- transitive. The class of semi-symmetric graphs was first introduced by Folkman in [36] (1967), where he constructed the first example of such graphs and posed a number of problems concerning these graphs. A first attempt to formulate the property of a semi-symmetric graph in pure group theoretical terms was made in [64]. The class of semi-symmetric graphs is too large to be classified, so it is reasonable to study such graphs with additional properties. On the one hand, when investigating graphs with specific properties, the cases of small valencies are always a good starting point. It is therefore not surprising that in recent years the emphasis on semi-symmetric graphs are mainly given to those with low valencies (see for instance [18, 25, 86]). On the other hand, it is natural to consider the possible actions of the automorphism groups on the vertex set and edge set. For example, primitivity and biprimitivity are usually chosen to be the preliminary assumptions. For semi-symmetric graphs it is natural to assume

7 1.3. Main results and the structure of this thesis biprimitivity. Up to date, regarding the problem of complete classification of vertex- primitive or vertex-biprimitive graphs with small valency, the results are very rare in the literature. In [72] Li, Lu and Maruˇsiˇcgave a complete classification of vertex-primitive arc-transitive graphs of valency 3 or 4. In another highly cited article [53], Ivanov and Iofinova classified vertex-biprimitive edge-transitive cubic graphs in 1985, which was based on the classification of amalgams of edge-transitive cubic graphs obtained by Goldschmidt [45]. In [26], Du and Xu classified semi-symmetric graphs of order 2pq.

1.3 Main results and the structure of this thesis

The overall picture of the thesis is as follows. It contains three parts, the first part consists of this introductory chapter and the next two chapters. The second part is composed by a single chapter, namely Chapter 4, which is the core of the thesis. The third part consists of Chapters 5–8, which contains different applications of the main result given in Chapter 4. In Chapter 2 and Chapter 3 we present a detailed setting for the work in this thesis. In particular, in Chapter 2 we introduce some basic notation while presenting a few well-known results on finite permutation groups. This chapter also contains an introduction on the structure of finite primitive groups, namely the O’Nan-Scott Theorem, a discussion of the outer automorphism groups of almost simple groups, and a brief description of the Aschbacher Theorem. In Chapter 3 we collect some specific notation while discussing a few preliminary results on graphs. The symmetries of graphs, and some general ways of constructing graphs in terms of group theoretical methods, are also included. Chapter 4 is the heart of this thesis. In this chapter we begin our investigation of finite primitive permutation groups with soluble stabilizers. It is shown that for such a primitive permutation group, the primitive type is of affine, almost simple, or product action. The product action type can be reduced to the almost simple type. For groups of the almost simple type, we first list all the soluble maximal subgroups when the socle of such a group is an alternating group, a sporadic simple group, or an exceptional simple group of Lie type. Then we put our main efforts on the case of classical groups of Lie type. Using the Aschbacher Theorem, we produce a list of the soluble maximal

Ci-subgroups of classical groups, based on which we were able to determine the soluble maximal subgroups of classical groups. Finally a complete classiﬁcation of such groups is obtained. The main result of this chapter may be stated as follows.

Theorem A. Let G be a ﬁnite primitive permutation group such that the point stabilizer

Gα is soluble. Then G is aﬃne, almost simple, or product action; and if G is almost

8 1.3. Main results and the structure of this thesis

simple, then the pair (G, Gα) is given in Theorem 4.1.1.

Chapter 5 contains the first application of Theorem A, where finite edge-primitive s-arc-transitive graphs, with s ≥ 4, are classified, solving a problem of Weiss (1999). The main result is as follows.

Theorem B. Let Γ be an edge-primitive and s-transitive graph with s ≥ 4. Then Γ is a known graph and is given in Theorem 5.1.1.

Chapter 6 is devoted to carrying out a general study on a class of locally primitive symmetric graphs, namely, the class of 2-path-transitive graphs. By using the local analysis, we obtain a lot of information on the local structures of 2-path-transitive graphs which are not 2-arc-transitive. We also present the relationship between the symmetries of 2-path-transitive graphs and their line graphs. It is shown that a graph is 2-path- but not 2-arc-transitive if and only if its line graph is half-transitive, based on which a class of half-transitive graphs is constructed from certain alternating groups. In Chapter 7 we investigate vertex-primitive and vertex-biprimitive 2-path-transitive graphs which are not 2-arc-transitive. A complete classiﬁcation is presented for such graphs. The classiﬁcation depends heavily on the main result of Chapter 4 and the local structure theorem on 2-path- but not 2-arc-transitive graphs given in Chapter 6. We state the main result of this chapter as follows.

Theorem C. Let Γ be a (G, 2)-path-transitive graph which is G-vertex-primitive or G-vertex-biprimitive. If Γ is not (G, 2)-arc-transitive, then the pair (Γ ,G) is given in Theorem 7.1.1 or Theorem 7.1.2, respectively.

The above classification result then allows us to construct some half-transitive graphs from certain almost simple groups. For the graphs of Theorem C, the numbers of non- isomorphic graphs are also determined in this chapter. Finally, in Chapter 8, as another important application of the main result of Chap- ter 4, we deal with vertex-biprimitive edge-transitive tetravalent graphs. A complete classification of such graphs is successfully obtained and may be simplified as follows.

Theorem D. Let Γ be a connected bipartite graph of valency 4. If Γ is biprimitive, edge-transitive but not vertex-transitive, then Γ is given in Theorem 8.1.1.

To end this chapter, we point out that the main results of this thesis completely answered Problem 1, Problem 3 and Problem 4, and partially but satisfactorily answered Problem 2.

9 Chapter 2

Preliminaries

In this chapter we collect basic notation, deﬁnitions, and preliminary results related to permutation groups and ﬁnite simple groups that will be used in subsequent chapters.

2.1 Permutation groups

Let Ω be a nonempty set. A bijection of Ω onto itself is called a permutation of Ω. Under composition of mappings, the set of all permutations of Ω forms a group, which is called the symmetric group on Ω, and is denoted as Sym(Ω). Any subgroup of Sym(Ω) is said to be a permutation group on Ω. In particular, if Ω is a finite set, say |Ω| = n, where n is a positive integer, then we say that Sym(Ω) is of degree n, and denote Sym(Ω) by Sn. Let g ∈ Sn. Then g is said to be a transposition if it interchanges two elements of Ω, and fixes the others. A permutation of Sn is called even if it is a product of an even number of transpositions. The set of even permutations of Sn forms an index two subgroup of Sn, which is called the alternating group, and is denoted by An. Let G be a group and Ω be a nonempty set. We refer to the elements of Ω as points. Suppose that for each point α ∈ Ω and each g ∈ G, there exists a unique point of Ω, which we denote by αg. Then this correspondence is a map from Ω × G into Ω. We say that this defines an action of G on Ω (or G acts on Ω) if the following two conditions are satisfied:

(i) α1 = α, for all α ∈ Ω, where 1 is the identity of G, and

g g g g (ii)( α 1 ) 2 = α 1 2 , for all α ∈ Ω and all g1, g2 ∈ G.

Suppose that G acts on Ω. Then a point α of Ω is said to be ﬁxed by an element g of G if αg = α. The elements of G which ﬁx all the points of Ω form a subgroup of G, called the kernel of this action, and the action is said to be faithful if the kernel is the identity subgroup.

10 2.1. Permutation groups

If G is a permutation group on Ω, that is, G ≤ Sym(Ω), then G acts naturally on Ω by permuting the points of Ω among themselves. Assume next that a group G acts on a nonempty set Ω. Then for each element g ∈ G, we can define a mapg ¯ of Ω into itself given byg ¯ : α → αg. It is easy to see thatg ¯ is a bijection, thus the map ρ : G → Sym(Ω) given by ρ : g → g¯ is well defined. By properties (i) and (ii), ρ is a group homomorphism. In general, for a group G and a nonempty set Ω, any homomorphism of G into Sym(Ω) is called a permutation representation of G on Ω. If ρ : G → Sym(Ω) is a (permutation) representation, then for any g ∈ G and α ∈ Ω, by defining αg: = (ρ(g))(α), we obtained an action of G on Ω. Therefore there is a correspondence between the group actions and permutation representations. It is clear that the kernel of this action is precisely kerρ, the kernel of ρ. The representation ρ is said to be faithful if kerρ = 1. As can be seen from the above discussion, permutation representations and actions are the different ways of describing the same situation. The following are a few well known permutation representations (actions) which will be used repeatedly in this thesis.

Example 2.1.1. Let G be a group. We simply take Ω: = G, and for each element g ∈ G, deﬁne two mapsg ˆ,g ˇ of G into itself as follows:

gˆ : x → xg, for all x ∈ G. gˇ : x → g−1x gˆ (ˇg) is called the right (left) multiplication induced by g. Let Gˆ = {gˆ | g ∈ G}, and Gˇ = {gˇ | g ∈ G}. Then both Gˆ and Gˇ are subgroups of Sym(G) which are isomorphic to G, and the maps ρ1 : g → gˆ, ρ2 : g → gˇ give two representations of G, called the right, left regular representation of G, respectively.

Example 2.1.2. Let G be a group, H ≤ G, and let [G:H] be the set of right cosets of H in G. We deﬁne an action of G on [G:H] by right multiplication, that is, (Hx)g: = Hxg, where g ∈ G, and Hx ∈ [G:H]. This action is called a coset action of G. It is clear that g ﬁxes Hx if and only if g ∈ x−1Hx, thus the kernel of this action is T x−1Hx. This x∈G kernel, which is the maximal normal subgroup of G contained in H, is called the core of

H in G, and is denoted by CoreG(H). If CoreG(H) = 1, then H is said to be core-free in G. Thus the action is faithful if and only if H is core-free.

Generally, let H1,H2,...,Hs be subgroups of G, and let

Ω = [G:H1] ∪ [G:H2] ∪ · · · ∪ [G:Hs].

Then the right multiplicative action of g on Ω is deﬁned as

g (Hix) : = Hixg for all g ∈ G.

11 2.2. Orbits, stabilizers, and transitivity

2.2 Orbits, stabilizers, and transitivity

Let G be a group acting on a set Ω. For a point α ∈ Ω, the set

αG: = {αg | α ∈ G} is called the orbit of α under G. The size |αG| is called the length of the orbit. The set of elements in G that ﬁx the point α constitute a subgroup of G, called the stabilizer of

α in G, and is denoted by Gα. Thus

g Gα = {g ∈ G | α = α}.

A group G acting on a set Ω is said to be transitive on Ω if it has only one orbit, that is, αG = Ω for all α ∈ G, otherwise it is called intransitive. If for each α ∈ Ω, we have Gα = 1, then G is said to be semi-regular on Ω. A semi-regular group is said to be regular if it is transitive. The following results are well known and can be found in [24].

Lemma 2.2.1. [24, Theorem 1.4 A] Let G be a group acting on a set Ω. For g ∈ G and α, β ∈ Ω, we have

(i) Two orbits αG and βG are either equal or disjoint.

−1 (ii) Gαg = g Gαg.

G G (iii) |α | = |[G:Gα]|. In particular, if G is ﬁnite, then |α ||Gα| = |G|.

Lemma 2.2.2. [24, Corollary1.4 A, Theorem 4.2 A] Suppose that G is transitive on the set Ω. Then

(i) The stabilizers Gα (α ∈ Ω) form a unique conjugacy class in G.

(ii) The centralizer CSym(Ω)(G) of G in Sym(Ω) is semi-regular.

(iii) If H is a transitive subgroup of G, then G = HGα, where α ∈ Ω.

(iv) If G is ﬁnite and abelian, then G is regular.

Suppose that G1 ≤ Sym(Ω1), and G2 ≤ Sym(Ω2). We say that G1 and G2 are permutationally isomorphic if there exists a bijection ϕ :Ω1 → Ω2 and a group isomorphism

ψ : G1 → G2 such that

g ψ(g) ϕ(α ) = ϕ(α) for all α ∈ Ω1 and g ∈ G1.

12 2.2. Orbits, stabilizers, and transitivity

In particular, if G1 = G2 = G, then the actions of G on Ω1 and Ω2 are said to be permutationally equivalent if there exists a bijection ϕ :Ω1 → Ω2 such that

g g ϕ(α ) = ϕ(α) for all α ∈ Ω1 and g ∈ G.

If the actions of a group on two diﬀerent sets are permutationally equivalent, then essentially we can identify the two actions. Any transitive permutation action of a group G on a set Ω may be identiﬁed with the action of G on the cosets of a point stabilizer.

Lemma 2.2.3. [24, Theorem 1.4 A] Let G be a group acting transitively on a set Ω. Then for each α ∈ Ω, the action of G on Ω is permutationally equivalent to the coset action of G on [G:Gα].

Let G be a group acting on a set Ω, and let ∆ be a nonempty subset of Ω. Then the set of elements of G that ﬁx ∆ pointwise is a subgroup of G, called the pointwise stabilizer of ∆ in G, and is denoted as G(∆). Thus

g G(∆) = {g ∈ G | δ = δ for all δ ∈ ∆}.

This extends the notation of a point stabilizer. If ∆ contains only two elements, say

∆ = α, β, then G(∆) is simply denoted as Gαβ. The setwise stabilizer of ∆ in G is deﬁned as g G∆: = {g ∈ G | ∆ = ∆}, which is also a subgroup of G, clearly G(∆) ¢ G∆. The subgroup G∆ has a natural action on ∆, and the kernel of this action is G(∆). The permutation group associated ∆ ∆ ∼ with this action is denoted by G∆, then G∆ = G∆/G(∆). For a positive integer k, we use Ω(k) to denote the set of all k-tuples of distinct points of Ω, and use Ω{k} to denote the set of all k-sets (that is, subsets of size k) of Ω. Then G induces a natural action both on Ω(k) and Ω{k}, given by

g g g (α1, . . . , αk) = (α1, . . . , αk) for all g ∈ G, and

g g g {α1, . . . , αk} = {α1, . . . , αk} for all g ∈ G,

respectively. We say G is k-transitive if G is transitive on Ω(k), and we say G is k- homogeneous if G is transitive on Ω{k}. Clearly k-transitive implies k-homogeneous. In this thesis some of our contents are closely related to the case of k = 2. As a consequence of the classiﬁcation of the ﬁnite simple group (CFSG), all 2-transitive permutation groups are known up to permutation isomorphism (see [10]). The structure of 2-homogeneous groups was given by Kantor [56], and we will introduce this result in Chapter 5.

13 2.3. Blocks, primitivity, and the O’Nan-Scott Theorem

2.3 Blocks, primitivity, and the O’Nan-Scott Theorem

Let Ω be a set. A partition B = {B1,B2,...,Bm} of Ω is a set of subsets of Ω such that Bi 6= Bj whenever i 6= j, and Ω = B1 ∪ B2 ∪ · · · ∪ Bm. Let G act on Ω. Then the g partition B is called G-invariant if for each Bi ∈ B and each g ∈ G, we have Bi ∈ B. A subset ∆ of Ω is said to be G-invariant if ∆g = ∆ for all g ∈ G, for example, every orbit of G is G-invariant. Assume that ∆ is G-invariant. Then the setwise stabilizer

G∆ is coincident with G. Thus the restriction of G on ∆ induces an action of G on ∆, ∆ the kernel of which is G(∆). Denoting the associated permutation group as G , then ∆ ∼ G = G/G(∆). Let G be a group acting transitively on a set Ω, and let ∆ be a nonempty subset of Ω. We say ∆ is a block for G if for each g ∈ G, either ∆g = ∆, or ∆g ∩ ∆ = ∅. The set Ω is clearly a block, and for each α ∈ Ω, the singleton {α} is also a block. These are called trivial blocks and any other block is called nontrivial. Assume that ∆ is a block for G. Then it is easy to show that for each g ∈ G, the set ∆g is also a block. Let Π := {∆g | g ∈ G}. Then Π is called the system of blocks containing ∆. Since G is transitive on Ω, the subsets in Π form a G-invariant partition of Ω. A transitive group G on Ω is said to be primitive if G has no nontrivial blocks; otherwise G is called imprimitive. Assume that G is imprimitive. Then there exists a system of nontrivial blocks which constitutes a G-invariant partition of Ω, called an imprimitive partition or a system of imprimitivity for G. A transitive group G on Ω is said to be biprimitive if Ω has a G-invariant partition Ω = Ω1 ∪ Ω2, and GΩ1 = GΩ2 is primitive on Ω1 and Ω2. Primitivity and biprimitivity play a central role in this thesis. First we present some basic properties of primitive groups, the proof of which can be found in [24].

Lemma 2.3.1. [24, Corollary 1.5 A, Theorem 1.6 A] Let G be a group acting transitively on a set Ω with |Ω| > 1. Then

(i) G is primitive if and only if for each α ∈ Ω the stabilizer Gα is a maximal subgroup of G.

(ii) If G is primitive on Ω, then any nontrivial normal subgroup of G is transitive on Ω.

(iii) If G is primitive and nonabelian, then CSym(Ω)(G) = 1.

We remark that the converse of the property (ii) is not true. A transitive permutation group G is said to be quasiprimitive if every nontrivial normal subgroup of G is also transitive.

14 2.3. Blocks, primitivity, and the O’Nan-Scott Theorem

Throughout this thesis, all our primitive and biprimitive groups are assumed to be ﬁnite. Combining Lemma 2.2.3 and Lemma 2.3.1, it can be seen that the study of the primitive actions of a ﬁnite group G is closely related to the maximal subgroups of G.

2.3.1 The O’Nan-Scott Theorem

As mentioned before, this work is closely related to finite primitive permutation groups. Therefore it is very important to get a better understanding on the structure of finite primitive groups. At the Santa Cruz Conference in finite groups in 1979, M. O’Nan and L. L. Scott independently proposed a classification scheme for finite primitive groups, which became “ a theorem of O’Nan and Scott” and finally “the O’Nan-Scott Theorem” (see [80], we remark that the original versions of O’Nan and Scott omitted the twisted wreath case). It turns out that the key to analyzing finite primitive groups is to study the socle. For a group G, the socle of G, denoted by soc(G), is defined to be the subgroup generated by the set of all minimal normal subgroups of G. The socle soc(G) is a (nontrivial) characteristic subgroup of G. It is known that the socle of a finite primitive group is a direct product of isomorphic simple groups. Nowadays, there are different versions of the O’Nan-Scott Theorem. However, according to the scheme, it is highly agreed that any satisfactory treatment of the O’Nan-Scott Theorem comprises two main parts:

(i) A list of types of primitive permutation groups;

(ii) An argument proving that each primitive group is permutationally isomorphic, or has the full normalizer of its socle permutationally isomorphic, to a unique group of the list.

For our purpose we will adopt the version given in [80] (see also [24]). Before presenting this version, we introduce a few types (or actions) of groups. First we take the concept of “semidirect product” of groups as a familiar one. Notice that, for a primitive group G ≤ Sym(Ω), N := soc(G) is transitive because it is nontrivial. Therefore by

Lemma 2.2.2, we have G = NGα for any α ∈ Ω. e (a) Affine groups: Let F = Fq be a field with q elements, where q = p , with p prime, and let V = F n be the n-dimensional vector space over F . Then for each vector v ∈ V , each linear transformation a ∈ GLn(q), and each field automorphism σ ∈ AutF , the affine semilinear transformation ta,v,σ : V → V is defined by

σ ta,v,σ : u → a(u ) + v.

The set of all aﬃne semilinear transformations is a subgroup of Sym(V ), denoted by

AΓLn(F ). The set of all the elements of the form ta,v,1 is a subgroup of AΓLn(F ), called

15 2.3. Blocks, primitivity, and the O’Nan-Scott Theorem

the affine group of dimension n over F , and is denoted by AGLn(F ). We define two more subgroups of the affine group AGLn(F ) as follows:

ASLn(q): = {ta,v,1 ∈ AGLn(F ) | deta = 1}, and

T : = {t1,v,1 | for all v ∈ V }.

The subgroup ASLn(q) is called the affine special linear group. The subgroup T acts regularly on V and the elements of T are called translations. It can be shown that the affine group AGLn(F ) is a 2-transitive subgroup of Sym(V ), and is therefore a primitive subgroup of Sym(V ). Furthermore, the socle soc(AGLn(F )) = T is an elementary abelian group which is isomorphic to the additive group of the filed F . The stabilizer of the zero vector in AGLn(F ) is isomorphic to the linear group GLn(q). Denoting the ∼ additive group of F as F˜, then we have AGLn(F ) = F˜:GLn(q). (b) Almost simple groups: A group G is said to be almost simple if T ¢G ≤ AutT for some finite nonabelian simple group T . In this case soc(G) = T . Assume that H is a maximal subgroup of G. Then the coset action of G on [G:H] is primitive.

(c) Product action: Let H ≤ Sym(∆) and K ≤ Sm, where ∆ is a ﬁnite set, and m > 1. We deﬁne an action of K on the direct product Hm given by

k (h1, h2, . . . , hm) = (h1k−1 , h2k−1 , . . . , hmk−1 ),

m where (h1, h2, . . . , hm) ∈ H , and k ∈ K. Then the wreath product of H by K is deﬁned to be the semi-direct product Hm:K with respect to this action. We denote this group by H o K. Furthermore, we can deﬁne an action of H o K on the set Ω: = ∆m given by

h −1 h −1 h −1 (h1,h2,...,hm,k) 1k 2k mk (δ1, δ2, . . . , δm) = (δ −1 , δ −1 , . . . , δ −1 ), 1k 2k mk

m where (δ1, δ2, . . . , δm) ∈ ∆ , and (h1, h2, . . . , hm, k) ∈ H o K. This action is called the product action of the wreath product. The following result can be found in [24].

Lemma 2.3.2. [24, Lemma 2.7 A] Let H ≤ Sym(∆), K ≤ Sm with m > 1, and H, K be nontrivial. Then H o K is primitive in the product action on ∆m if and only if

(i) H acts primitively but not regularly on ∆ and

(ii) K is a transitive subgroup of Sm.

(d) Groups of diagonal type: Let T be a nonabelian simple group and let H = T m, with m > 1. The subgroup

D: = {(t, t, . . . , t) | t ∈ T } =∼ T

16 2.3. Blocks, primitivity, and the O’Nan-Scott Theorem

m is called the diagonal subgroup of T . Let Ω = [H : D]. For any (t1, t2, . . . , tm)D ∈ Ω 0 0 0 and (t1, t2, . . . , tm) ∈ H, we deﬁne an action of H on Ω given by

0 0 0 (t1,t2,...,tm) 0 0 0 ((t1, t2, . . . , tm)D) = (t1t1, t2t2, . . . , tmtm)D.

Then it is easy to show that this action is faithful and transitive. Therefore H <

Sym(Ω). Let N: = NSym(Ω)(H). Then by [24, Lemma 4.5 B], and by slightly abusing ∼ the notation, we have H < T o Sm ≤ N = (T o Sm).Out(T ). We say that a subgroup G of Sym(Ω) is of diagonal type if H ≤ G ≤ N. With additional conditions, the group G can be made into a primitive group (see [24, Theorem 4.5 A] for details). For our purpose we only need the following result.

Lemma 2.3.3. With the notation above, if G is of diagonal type, then for any α ∈ Ω, the point stabilizer Gα is insoluble.

Proof. Since H = T m is transitive on Ω, so is G. Hence we may take α = D ∈ Ω. Then ∼ Gα ≥ Hα = D = T , therefore Gα is insoluble.

(e) Twisted wreath product: Let T and K be groups, and let L < K. Assume

that the index |[K:L]| = m > 1, and {y1, y2, . . . , ym} is a set of right coset representatives for L in K. Assume further that there exists a homomorphism ϕ : L → AutT . Let m H: = T . For k ∈ K, assume that yik ∈ Lyj. Then there exists an element li ∈ L such π that yik = liyj. Hence k induces a permutation πk ∈ Sm such that i k = j. Now with respect to k, we deﬁne a map k˜ : H → H given by

˜ 0 0 0 k :(t1, t2, . . . , tm) → (t1, t2, . . . , tm),

0 ϕ(li) where (t1, t2, . . . , tm) ∈ H, and tj = ti . Since ϕ is a homomorphism from L to AutT , it is readily veriﬁed that k˜ is an automorphism of H. Thus we can deﬁne the semidirect product G = H:K, which is called the twisted wreath product with respect to the triple

(T, K, ϕ), and is denoted as T oϕ K. Under some circumstances, the group T , K and the map ϕ can be chosen such that one can define a primitive action for the twisted wreath product T oϕ K (see [24, Lemma 4.7 A] for details). But there is no known necessary and sufficient conditions on T , K and ϕ such that T oϕ K is a primitive group. We have seen that the socle of a finite primitive group is a direct product of isomorphic simple groups. Let G ≤ Sym(Ω) be a finite primitive group with a regular nonabelian socle of the form T m (T simple). Then it is known that G is isomorphic to a twisted wreath product T oϕ K, where |[K:L]| = m, |Ω| = |T |m, and ϕ : L → AutT has the property that Imϕ ≥ InnT , see [80]. For α ∈ Ω, since Gα = K, we obtain the following result immediately

17 2.4. More on almost simple groups

Lemma 2.3.4. If G is a primitive group isomorphic to a twisted wreath product, then

Gα is insoluble.

Now we can give the O’Nan-Scott Theorem. For a vector space V , recall that a subgroup H of GL(V ) is said to be irreducible if none of the nontrivial subspaces of V is ﬁxed by H.

Theorem 2.3.5. (O’Nan-Scott Theorem) Let G ≤ Sym(Ω) be a ﬁnite primitive group of degree n, and let N be the socle of G. Then one of the following holds.

∼ m (i) G is aﬃne type and N = Zp is a regular elementary abelian p-group, where m n = p , with p prime. G can be embedded in the aﬃne group AGLm(p), with N

identiﬁed with the subgroup of all translations. Furthermore, Gα is isomorphic to

an irreducible subgroup of GLm(p).

(ii) G is almost simple and T ¢ G ≤ AutT for some ﬁnite nonabelian simple group T .

(iii) G is diagonal type, N = T m with m ≥ 2, n = |T |m−1, and G is a subgroup of a wreath product with the diagonal action.

(iv) G is product type and N = T m with m = st and t > 1. There is a primitive group H, which is almost simple or diagonal type, such that G is isomorphic to a

subgroup of ≤ H o St, with the product action.

(v) G is twisted wreath product type.

Remark Assume that G ≤ Sym(Ω) is of type (iv), and for α ∈ Ω, assume that Gα is soluble. Then by Lemma 2.3.3, the primitive type of H is not diagonal.

2.4 More on almost simple groups

Let G be an almost simple group. Then soc(G) = T , and T ¢ G ≤ AutT for some finite nonabelian simple group T . Furthermore, AutT = InnT.OutT =∼ T.OutT . By Schreier’s theorem (also known as Schreier’s conjecture), the outer automorphism group OutT is soluble. According to the classification of the finite simple groups (henceforth referred to as CFSG), the finite almost simple groups fall into four classes, namely, the alternating groups An and symmetric groups Sn (n ≥ 5); the sporadic simple groups and their automorphism groups; the exceptional almost simple groups of Lie type; and the classical almost simple groups of Lie type. In this section, we list all the socles of the finite almost simple groups and all the outer automorphism groups of the socles for later use.

(1) Alternating groups An (n ≥ 5). The full automorphism group of An is Sn, 2 except for n = 6, in which case AutA6 = A6.Z2.

18 2.4. More on almost simple groups

(2) Sporadic simple groups. There are 26 sporadic simple groups, of which the following 14 groups

M11, M23, M24, Co1, Co2, Co3, Ru, Ly, J1, J4, Fi23, Th, B and M have trivial outer automorphism groups, and the remaining 12 groups.

0 0 M12, M22, J2, Suz, HS, McL, Fi22, O N, HN, Fi24, He and J3 have outer automorphism groups of order 2. (3) Exceptional simple groups of Lie type. The exceptional simple groups of Lie type and the orders of their outer automorphism groups are listed in the following table (where q = pe).

2 2 2 T F4(q) E6(q) E7(q) E8(q) B2(q) G2(q) F4(q) q = 22m+1 q = 32m+1 q = 22m+1 |OutT | (2, p)e 2(3, q − 1)e (2, q − 1)e e e e e 3 2 T G2(q) D4(q) E6(q) |OutT | e if p 6= 3 3e 2(3, q + 1)e 2e if p = 3

(4) Classical simple groups of Lie type. The classical groups of Lie type fall into four types: the linear groups, the symplectic groups, the unitary groups and the orthogonal groups of three families, as shown in the following

PSLn(q), PSUn(q), PSpn(q)(n even) , ± PΩn (q)(n even), PΩn(q)(nq odd).

For this class of simple groups we need more detailed information about their outer automorphism groups. By a result of Steinberg (see [11, Chapter 12]), the outer automorphism groups of the groups of Lie type have a uniform description in terms of so-called diagonal, ﬁeld and graph automorphisms. Any automorphism group can be expressed as a product of an inner, a diagonal, a ﬁeld, and a graph automorphism. Let q = pe, with p prime. The following information can be found in [62].

1) Assume that T = PSLn(q) is a linear group, where n ≥ 2, and (n, q) ∈/ {(2, 2), (2, 3)}. Then we have:

∼ Z(n,q−1).Ze.Z2, if n ≥ 3, Out(PSLn(q)) = Z(n,q−1) × Ze, if n = 2.

2) Assume that T = PSpn(q) is a symplectic group, with n even, and (n, q) ∈/ {(2, 2), (2, 3)}. Then the following hold:

19 2.5. The subgroup structure of the classical groups

∼ Ze, if n 6= 4, and q is even, Out(PSpn(q)) = Z2 × Ze, if q is odd. e If T = PSp4(2 ), then by [11, Chapter 12], T admits a graph automorphism of order 2, and Out(T ) = Z2e.

3) Assume that T = PSUn(q) is a unitary group, where n ≥ 2, and (n, q) ∈/ {(2, 2), (2, 3), (3, 2)}. Then T has no graph automorphism, and

∼ Out(PSUn(q)) = Z(n,q+1) : Z2e.

4) Assume that T = PΩn(q) is an orthogonal group of odd dimension. Then T has no graph automorphism, and

∼ Out(PΩn(q)) = Z2 × Ze.

− 5) Assume that n is even, and T = PΩn (q) is an orthogonal group of the minus type. Then we have

  Z2e, if q is even, − ∼ Out(PΩn (q)) = Z2 × Z2e, if q is odd and n(q − 1)/4 is even,  D8 × Ze, if q and n(q − 1)/4 are odd. + 6) Assume that n 6= 8 is even, and T = PΩn (q) is an orthogonal group of the plus type. Then

  Z2 × Ze, if q is even, + ∼ Out(PΩn (q)) = Z2 × Z2 × Ze, if q is odd and n(q − 1)/4 is even,  D8 × Ze, if q and n(q − 1)/4 are odd.

+ Assume that T = PΩ8 (q). T admits a graph automorphism, known as the triality automorphism, of order 3. Further, we have

+ ∼ S3 × Ze, if q is even, Out(PΩ8 (q)) = S4 × Ze, if q is odd.

2.5 The subgroup structure of the classical groups

We will determine the soluble maximal subgroups of almost simple groups in Chapter 4, based on Aschbacher’s classification of the maximal subgroups of classical groups. Aschbacher’s seminal contribution has been very influential both on the way problems concerning classical groups are analysed, and on the way results about such groups are presented. The classification divides the maximal subgroups of classical groups into nine

20 2.5. The subgroup structure of the classical groups special classes (see [1]). These classes are usually defined in terms of some geometrical property associated with the action on the underlying vector space. Subgroups in these classes are therefore called geometric subgroups. An earlier version of the Aschbacher Theorem was proved in [28] by Dynkin in 1952, thus the result is also known as the Aschbacher-Dynkin Theorem. Let G be an almost simple classical group of Lie type f defined on an n-dimensional vector space V over Fq, where q = p with p prime. A rough description of this classification is given as follows:

C1 the stabilizers of a non-singular subspace, or the stabilizers of a pair of subspaces {U, W } such that U < W and dim(U) + dim(W ) = n (these subgroups act re- ducibly on V , and are known as the maximal parabolic subgroups);

C2 the stabilizers of a partition of V into isometric non-singular or totally singular subspaces (these subgroups act irreducibly but imprimitively on V );

C3 the stabilizers of non-singular subspaces deﬁned on an extension ﬁeld of prime degree;

C4 the stabilizers of a decomposition of V as a tensor product of two non-isometric subspaces;

C5 the stabilizers of a proper subﬁeld of Fq, of prime degree;

C6 the normalizers of certain symplectic-type r-groups;

C7 the stabilizers of a decomposition of V as a tensor product of isometric subspaces;

C8 a classical subgroup.

C9 a subgroup which is almost quasisimple and absolutely irreducible on the natural module.

The detailed deﬁnitions of the classes Ci can be found in [62, Chapter 4]. The main result of Aschbacher’s paper [1] states that, roughly speaking, any subgroup of G is contained in a member of Ci. We will give a special version of the Aschbacher Theorem in Chapter 4.

21 Chapter 3

Graphs

All the graphs in this thesis are assumed to be ﬁnite, simple, and connected. In this chapter we present basic notation and properties on graphs, including the basic concepts of graphs, the symmetries of graphs, and a few of the most common graph constructions.

3.1 Basic concepts of graphs

Graphs come in two types: directed graphs (or digraphs) and undirected graphs. We use the term graph to refer to undirected graphs. The following are the definitions. Let V be a nonempty set, the elements of which are called vertices.A digraph Γ is a pair (V,E) where E is a subset of V × V . The elements of E are called edges. The digraph Γ is said to be finite if the vertex set V is finite. Let e = {α, β} ∈ E be an edge. Then e is said to join α to β, and β is adjacent to α. We also say that α and e (or β and e) are incident. If {α, β} ∈ E whenever {β, α} ∈ E, then the digraph is called undirected. A digraph is called simple if there exist no edges of the form {α, α} and has no multiple edges between any two vertices. By a graph we mean a digraph which is simple and undirected. Mainly we will consider undirected graphs in this thesis. Usually we use Γ ,Σ,... to denote graphs. Let Γ be a graph. Sometimes the vertex set and edge set of Γ are also denoted by V Γ and EΓ , respectively. The size |V Γ | is called the order of Γ . For an edge {α, β} ∈ EΓ , the ordered sets (α, β) and (β, α) are called the arcs corresponding to this edge. The set of all the arcs of Γ is denoted as AΓ . For a vertex α ∈ Γ , denote by Γ (α) the set of vertices of Γ which are adjacent to α (the set Γ (α) may be empty), called the neighborhood of α. The size |Γ (α)| is called the valency of α. If |Γ (α)| = 0, then α is called an isolated vertex. If all vertices of Γ have the same valency, say k, then Γ is called regular of valency k. In particular, a regular graph of valency 3 is also called a trivalent graph or cubic graph, and a regular graph of valency 4 is called a tetravalent graph. Unless otherwise stated, all graphs in this thesis are assumed to be regular.

22 3.2. Symmetries of graphs

Let Γ be a digraph. A sequence of s + 1 distinct vertices α0, α1, . . . , αs is called a

directed path of length s, or simply a s-dipath if αi is adjacent to αi+1 for all 0 ≤ i < s.

Identifying the two dipaths α0, α1, . . . , αs and αs, αs−1, . . . , α0 gives rise to an s-path. The graph Γ is called connected if for any two vertices α, β ∈ V Γ , there exists a path from α to β. The distance between two vertices is deﬁned to be the length of the shortest path between them. A graph Γ is called bipartite if the vertex set V Γ can be divided into two disjoint parts U and W , such that every edge connects a vertex in U to a vertex in W . The following are some special graphs.

(1) The complete graph Kn: a graph of order n in which any two vertices are adjacent.

(2) The cycle Cn: a graph of order n, with V Cn = {α1, α2, . . . , αn}, and ECn =

{{αi, αi+1} | 1 ≤ i ≤ n, with αn+1: = α1}.

(3) The complete bipartite graph Km,n: a bipartite graph with parts U and W , such that |U| = m, |W | = n, and each vertex of U is adjacent to each vertex of W .

(4) The matching n.K2: a graph with n isolated edges.

Let Γ = (V,E) be a graph. The line graph (also known as edge graph) L(Γ ) of Γ is

deﬁned as follows: the vertex set of L(Γ ) is E, and two vertices e1 and e2 of L(Γ ) are adjacent if and only if they are incident with some vertex α ∈ V .

3.2 Symmetries of graphs

Let Γ = (V,E) be a graph. An element g of Sym(V ) is called an automorphism of Γ if g preserves the adjacency of Γ , that is, for any {α, β} ∈ E, we have {αg, βg} ∈ E. The set of all the automorphisms of Γ forms a subgroup of Sym(V ), called the full automorphisms group of Γ , and is denoted as AutΓ .

The following results are well-known: Aut(Kn) = Sn; Aut(Cn) = D2n (the dihedral

group of order 2n); Aut(Kn,n) = Sn o S2; and Aut(Km,n) = Sm × Sn (where m 6= n). Let G ≤ AutΓ . If G acts transitively on V Γ , then Γ is said to be G-vertex-transitive, or simply vertex-transitive. It is clear that G induces a natural action both on the edge set EΓ and on the arc set AΓ . We say that Γ is G-edge-transitive (or simply edge- transitive) if G is transitive on EΓ , and we say that Γ is G-arc-transitive (or simply arc-transitive) if G is transitive on AΓ . An arc-transitive graph is also known as a symmetric graph. It is readily veriﬁed that an arc-transitive graph without isolated vertices is vertex transitive and edge-transitive, but the converse is not true. A graph is said to be half-transitive if it is vertex-transitive, edge-transitive, but not arc-transitive,

23 3.2. Symmetries of graphs while a graph is said to be G-semi-symmetric if it is G-edge-transitive but not G-vertex- transitive.

Lemma 3.2.1. If Γ be a G-semi-symmetric graph without isolated vertices, then Γ is bipartite.

Proof. Assume that {α, β} ∈ EΓ , and let U: = αG, W : = βG. Since Γ is connected and G-edge-transitive, we have V Γ = U ∪ W . Since Γ is vertex-intransitive, we have U ∩ W = ∅. It follows that Γ is bipartite with parts U and W .

A graph Γ is called vertex-primitive (resp, edge-primitive) if for some G ≤ AutΓ , G acts primitively on V Γ (resp, on EΓ ). The graph Γ is called vertex-biprimitive if it is bipartite, the automorphism group AutΓ is imprimitive on V Γ , and the subgroup of AutΓ which ﬁxes each of the two parts setwise acts primitively on each part. Let Γ be a graph and s a positive integer. An s-arc of Γ is an ordered (s + 1)-tuple of vertices (α0, α1, . . . , αs) such that {αi−1, αi} ∈ EΓ for 1 ≤ i ≤ s, and αi−1 6= αi for 1 ≤ i ≤ s − 1. For G ≤ AutΓ , the graph Γ is called (G, s)-arc-transitive if G acts transitively on the set of s-arcs. If in addition, G is not transitive on the set of (s + 1)-arcs, then Γ is called (G, s)-transitive. In particular, if G =Aut(Γ ), then a (G, s)-arc-transitive graph or a (G, s)-transitive graph is simply called an s-transitive graph. An s-transitive graph with s ≥ 2 is usually called a highly symmetrical graph. It is widely believed that highly symmetrical graphs are rare.

Let G ≤ AutΓ . Then for each α ∈ V Γ , the point stabilizer Gα induces an action on the neighborhood Γ (α), which is called the local action. We denote the kernel of [1] Γ(α) Γ(α) this action as Gα , and denote the induced permutation group by Gα . Then Gα =∼ [1] [1] [1] [1] [1] G/Gα . For αi ∈ V Γ , where 1 ≤ i ≤ n, we also deﬁne Gα1α2...αn : = Gα1 ∩Gα2 ∩· · ·∩Gαn .

For a positive integer k, the set Γk(α) is defined to be the set of vertices with distance at most k from α, thus Γ (α) = Γ1(α)\{α}. The pointwise-stabilizer of Γk(α) is denoted [k] by Gα . Γ(α) A graph Γ is called G-locally-primitive if for each α ∈ V Γ , the group Gα is primitive. All the graphs that we will study in subsequent chapters are locally-primitive. In recent years, the study of locally-primitive graphs has received considerable attention in the literature. The attention is mainly focused on two different directions, namely the “global-action analysis” and the “local-action analysis”. Global-action analysis aims to characterize the structure of an automorphism group G and its actions on the graph and various quotient graphs. In this direction a framework has been established by Cheryl E. Praeger (see [95, 96]), indicating that the case where G is an almost simple group is fundamental. Local-action analysis aims to find an upper-bound and the possible

24 3.2. Symmetries of graphs

structures for the stabilizer Gα of a vertex α. In this work both directions will be stressed. Edge transitivity and arc transitivity can be described by local actions, as shown in the following well-known result. Lemma 3.2.2. Let Γ be a connected graph, and let G ≤ AutΓ . Then Γ(α) (i) Γ is G-edge-transitive if and only if for each α ∈ V Γ , the group Gα is transitive.

Γ(α) (ii) Γ is G-arc-transitive if and only if Γ is G-vertex-transitive, and the group Gα is transitive for some α ∈ V Γ.

Proof. (i) Assume that for each α ∈ V Γ , the group Gα is transitive on Γ (α). Let {α, β} and {γ, δ} be two edges. Since Γ is connected, there exists a path connecting α and δ,

say {α, β} := {α1, β1}, {α2, β2},..., {αk, βk} := {γ, δ}, such that {αi, βi} ∩ {αi+1, βi+1}

is a singleton, say {βi}. As Gβi is transitive on Γ (βi), we can ﬁnd an element gi such g that {αi, βi} i = {αi+1, βi+1}. Therefore the element g1g2 . . . gk−1 sends the edge {α, β} to {γ, δ}. That is, Γ is G-edge-transitive. The other direction is clear. (ii) Assume that Γ is G-arc-transitive. Then it is G-vertex-transitive. Furthermore, for any β, γ ∈ Γ (α), there exists an element g which sends the arc (α, β) to the arc g g (α, γ), thus α = α, and β = γ. Hence Gα is transitive on Γ (α). Conversely, assume Γ(α) that Γ is G-vertex-transitive, and for some α ∈ V Γ , Gα is transitive. Then we ﬁx an g arc (α, β), and for any arc (α1, β1), there exists an element g ∈ G such that α1 = α, so g g h that β1 , β ∈ Γ (α). Therefore, there exists an element h ∈ Gα such that (β1 ) = β, and gh hence (α1, β1) = (α, β). It follows that Γ is G-arc-transitive.

There are a few occasions in this thesis where we need the concept of a normal quotient graph. Let G ≤ AutΓ and N ¢ G. Then the (normal) quotient graph ΓN of Γ with respect to N is deﬁned to be the graph with vertex set {αN | α ∈ V Γ }, the set of N-orbits, such that αN and βN are adjacent if and only if there exist x ∈ αN , y ∈ βN ,

and x, y are adjacent in Γ . The normal quotient ΓN of Γ inherits a lot of properties

of Γ . If Γ and ΓN have the same valency, then Γ is called a cover of ΓN ; while if the

valency of Γ is a multiple of the valency of ΓN , then Γ is called a multicover of ΓN . For edge-transitive graph we have Lemma 3.2.3. [74, Lemma 11.3.1] Let Γ be a connected G-edge-transitive graph, and

let N ¡ G be intransitive on V Γ . Then Γ is a multicover of ΓN . If |Γ (α) ∩ B| = 0 or 1 for any N-orbit B and α ∈ V Γ \B, then one of the following holds:

(i) Γ is a cover of ΓN .

(ii) N is semiregular on V Γ.

(iii) G/N ≤ AutΓN .

25 3.3. General constructions of graphs

3.3 General constructions of graphs

In algebraic graph theory, graphs are usually constructed in three different ways: the group theoretical way, the combinatorial way, and constructing a new graph from a given graph. Our approach is mainly the first one. In this section we introduce several common ways of constructing graphs, and discuss simple properties of these graphs. We remark that, with respect to the group theoretical methods, perhaps the most common graphs are the Cayley graphs. This type of graph is less interesting here, therefore we do not include them. The first type of construction produces the coset graphs. 1. Coset graphs. Definition 3.3.1. Let G be a group, H a core-free subgroup of G, and S a nonempty subset of G. Define the coset graph of G with respect to H and S to be the digraph with vertex set V = [G:H], such that for any Hx, Hy ∈ V , Hx is joined to Hy if and only if y−1x ∈ HSH. This digraph is denoted by Cos(G, H, HSH).

The coset graphs were introduced by Sabidussi [103], who showed that any vertex- transitive graph is isomorphic to a coset graph. First we collect some basic properties of such graphs, much of this is folklore. Lemma 3.3.2. Let Γ = Cos(G, H, HgH). Then (i) G ≤ AutΓ , and Γ is G-vertex-transitive.

(ii) Γ is undirected if and only if HSH = HS−1H.

(iii) Γ is connected if and only if hH,Si = G.

Proof. (i) This is clear. (ii) Assume that Γ is undirected. Then for any s ∈ S, we have {H, Hs} ∈ EΓ , thus {Hs, H} ∈ EΓ , therefore s−1 ∈ HSH. It follows that HSH = HS−1H. Conversely, assume that HSH = HS−1H, and {Hx, Hy} ∈ EΓ . Then y−1x ∈ HSH, so xy−1 ∈ (HSH)−1 = HS−1H = HSH. Therefore {Hy, Hx} ∈ EΓ , that is, Γ is undirected. (iii) Assume that Γ is connected. Then for any x ∈ G, there exists a path from the vertex H to the vertex Hx, say H = Hx0, Hx1, . . . , Hxt, Hxt+1, where xt+1 = x. Since −1 xixi−1 ∈ HSH, we conclude that x ∈ hH,Si, or hH,Si = G. Conversely, assume that 0 0 0 hH,Si = G. Then for any y ∈ G, suppose that y = htsththt−1st−1ht−1 . . . h1s1h1. Then 0 0 0 H, Hx1, . . . , Hxt, where xi = hisihihi−1si−1hi−1 . . . h1s1h1, is a path from H to Hy. It follows that Γ is connected.

For our purpose we are mainly interested in the cases where the graphs are arc- transitive. In such cases the set S may be chosen to be a singleton. Some of the proofs of the following results can be found in [83].

26 3.3. General constructions of graphs

Lemma 3.3.3. Let Γ = Cos(G, H, HgH), and let α be the vertex H. Then Γ (α) =

{Hgh | h ∈ H}, and Γ is G-arc-transitive. Moreover, the action of Gα on Γ (α) is equivalent to the coset action of H on [H:H ∩ Hg]. In particular, the valency of Γ is |H|/|H ∩ Hg|.

−1 Proof. It is clear that Gα = H. For any Hgh ∈ HgH, where h ∈ H, since ghh ∈ HgH, we have that Hgh is connected to Hh = H, and thus HgH ⊆ Γ (α). Conversely, assume

that Hy ∈ Γ (α). Then y ∈ HgH, say y = h1gh2. Thus Hy = Hgh2 ∈ HgH, so

that Γ (α) = HgH. Note that Γ is vertex-transitive, and Gα = H is transitive on Γ (α) = HgH. By Lemma 3.2.2, Γ is G-arc-transitive. Write L = H ∩ Hg, and deﬁne a map φ : Γ (α) → [H:H ∩ Hg] given by

φ : Hgh → Lh, for all h ∈ H.

Then it is easily shown that φ is a permutation equivalence from the action of Gα on g Γ (α) to the coset action of H on [H:H ∩ H ].

Lemma 3.3.4. Let Γ be a digraph and G ≤ AutΓ . Then Γ is G-arc-transitive if and only if there exists an element f ∈ G such that f 2 ∈ H, and Γ =∼ Cos(G, H, HfH).

Proof. Since Γ is G-arc-transitive, it is G-vertex-transitive. Hence Γ can be written as a coset graph: Γ =∼ Cos(G, H, HSH). Let α = H, and let g ∈ S. Then β: = Hg ∈ Γ (α). H Since Gα = H is transitive on Γ (α), we have that Γ (α) = β = HgH. It follows that HSH = HgH, so Γ =∼ Cos(G, H, HgH). Let f be the element that interchanges the arc 2 ∼ (H, Hg) with the arc (Hg, H). Then f ∈ H, and Γ = Cos(G, H, HfH).

The following simple fact is useful for determining the automorphism group of a coset graph.

Lemma 3.3.5. If Γ = Cos(G, H, HgH) and σ ∈ Aut(G), then Γ =∼ Cos(G, Hσ,HσgσHσ).

Proof. Deﬁne a map ϕ : Γ → Cos(G, Hσ,HσgσHσ) given by

ϕ : Hx → Hσxσ.

Then it is readily veriﬁed that ϕ is an automorphism.

Let Γ = Cos(G, H, HgH), and let Aut(G, H) = {σ ∈ Aut(G) | Hσ = H}. Some elements of Aut(G, H) induce automorphisms of Γ .

Lemma 3.3.6. Suppose that σ ∈ Aut(G, H). Then Γ = Cos(G, H, HgH) is isomorphic to Σ = Cos(G, H, HgσH). Moreover, σ induces an automorphism of Γ if and only if HgH = HgσH.

27 3.3. General constructions of graphs

Proof. Since σ ∈ Aut(G, H), by Lemma 3.3.5, we have that Γ is isomorphic to Σ. Thus σ induces an automorphism of Γ if and only if Γ = Σ, and this in turn is true if and σ only if HgH = Hg H.

2. Orbital graphs and double covers. The orbital graphs are well known so we do not give a formal deﬁnition. Let G be a transitive permutation group on a set Ω. Then G induces a natural action on the Cartesian product Ω × Ω. For any (α, β) ∈ Ω × Ω, the orbit ∆: = (α, β)G is called an orbital, and the orbital ∆∗: = (β, α)G is called the paired orbital of ∆. If ∆ = ∆∗, then we say that ∆ is self-paired. The digraph Σ: = (Ω, ∆) with vertex set Ω and edge set ∆ is called the orbital graph of G. Using Σ, we can deﬁne a new digraph Σ0: = (Ω, ∆∪∆∗), which is undirected, and is G-vertex-transitive and G-edge-transitive. Furthermore, Σ0 is G-arc-transitive if and only if ∆ is self-paired. Another type of construction is by building the standard double covers of a given graph.

Deﬁnition 3.3.7. Let Γ be a directed or undirected graph. Then the standard double cover of Γ is deﬁned to be the undirected graph, which we denoted by Γ˜ , such that V Γ˜ = V Γ × {1, 2}, and two vertices (x, i) and (y, j) are adjacent if and only if i 6= j, and {x, y} ∈ EΓ .

The new graph Γ˜ is bipartite with parts V Γ × {i}, where i = 1 or 2. The properties of Γ˜ are closely related to those of Γ . For example, Γ and Γ˜ have the same valency. Let G ≤ AutΓ . Then for each g ∈ G, the mapg ¯ : V Γ˜ → V Γ˜ given byg ¯ :(x, i) → (xg, i) is an automorphism of Γ˜ . Therefore the group G may be viewed as a subgroup of AutΓ˜ . It is clear that G ﬁxes each bipartite half V Γ × {i}, and for any α ∈ V Γ , we have Gα = G(α,i). Furthermore, we have

Lemma 3.3.8. Let Γ be an undirected graph, and Γ˜ the standard double cover of Γ . Then

(i) Γ˜ is connected if and only if Γ is connected and not bipartite.

(ii) If Γ is G-locally primitive, then Γ˜ is G-locally primitive.

(iii) If Γ is G-vertex-primitive, then Γ˜ is G-vertex-biprimitive.

Proof. (i) and (ii) follow from [41, Lemma 3,3], while (iii) is obvious.

Some edge-transitive graphs occur as the standard double cover of certain arc- transitive graphs, as the following result shows.

28 3.3. General constructions of graphs

Lemma 3.3.9. [41, Lemma 3,4] Let Γ be a G-edge-transitive graph of valency at least two with two vertex orbits ∆1 and ∆2 such that all vertex stabilizers are conjugate. Then

Γ is the standard double cover of an orbital digraph Σ of G on ∆1, and the bijection σ given by σ :(x, i) → (x, 3 − i) is an automorphism for Γ if and only if Σ is self-paired.

3. Double coset graphs. Usually an edge-transitive graph can be expressed as a double coset graph deﬁned as follows.

Deﬁnition 3.3.10. Let G be a group, and let L, R are subgroups of G such that L ∩ R is core-free in G. Deﬁne the bipartite graph

Γ = Cos(G, L, R) with the vertex set ∆1 ∪ ∆2, where ∆1: = {Lx | x ∈ G} and ∆2: = {Ry | y ∈ G}, such that Lx is adjacent to Ry if and only if yx−1 ∈ RL.

Following [55], we refer to (L, R, L ∩ L) as the associated amalgam. To study edge- transitive graphs, it is essential to investigate their amalgams. This “amalgam method” was first introduced by Goldschmidt in [45], and later was developed into an important new theory (see [21, 55]). Note that the condition yx−1 ∈ RL holds if and only if Lx∩Ry 6= ∅, and the above definition coincides with Tits’ construction of flag-transitive incidence structures (see [22]). Double coset graphs are also introduced in [31] and [41] to study locally s-arc-transitive graphs. Let Γ = Cos(G, L, R), and g ∈ G. Define a mapg ˆ : V Γ → V Γ given by

gˆ : Lx → Lxg, Rx → Rxg, for any x ∈ G.

Then it is readily seen thatg ˆ preserves the adjacency of Γ . Thusg ˆ ∈ AutΓ . As G =∼ Gˆ: = {gˆ | g ∈ G}, we may consider G to be a subgroup of AutΓ . We list some basic properties of double coset graphs.

Lemma 3.3.11. [41, Lemma 3,7] Let Γ = Cos(G, L, R) be a double coset graph. Then Γ is bipartite, and the following hold:

(i) Γ is connected if and only if hL, Ri = G.

(ii) For α = L, β = R ∈ V Γ , we have |Γ (α) = |L:L ∩ R|, and |Γ (β) = |R:L ∩ R|.

(iii) Γ is G-edge-transitive and G-vertex-intransitive.

Conversely, let Γ be a G-edge-transitive and G-vertex-intransitive graph. Then Γ =∼

Cos(G, Gα,Gβ), where {α, β} ∈ EΓ.

The coset graphs and the double coset graphs are related in the following way.

29 3.3. General constructions of graphs

Lemma 3.3.12. Let G be a group and L a subgroup. Assume that g ∈ G is such that L∩ Lg is core-free. Then Γ = Cos(G, L, Lg) is the standard double cover of Cos(G, L, LgL), 2 x g and Γ is arc-transitive if there exists x ∈ G such that x ∈ NG(L), and L = L .

Proof. By [41, Lemma 3.8], Γ = Cos(G, L, Lg) is the standard double cover of Cos(G, L, LgL). 2 x g g If there exists x ∈ G such that x ∈ NG(L) and L = L , then Γ = Cos(G, L, L ) = Cos(G, L, Lx). Let α, β be the vertices of Γ corresponding to L and Lg, respective- 2 ly. Then (α, β) is an arc, and as x ∈ NG(L), x induces an automorphism of Γ that x interchanges α and β. So Cos(G, L, L ) is arc-transitive.

30 Chapter 4

Finite primitive groups with soluble stabilizers.

This chapter, which is devoted to the study of ﬁnite primitive groups with soluble stabilizers, is the core of this thesis. By the O’Nan-Scott Theorem, the primitive types of such groups are aﬃne, almost simple, and product action (see Chapter 2). In this thesis we will focus our attention mainly on the almost simple type (it will be seen that the product action type can be reduced to the almost simple type).

4.1 The main result

A permutation group acting on a set is primitive if and only if for any point in the set, the stabilizer of which is a maximal subgroup of the permutation group. Thus the problem of studying finite primitive groups with soluble stabilizers is reduced to study the soluble maximal subgroups of primitive groups. It was claimed in [62] that all soluble maximal subgroups of almost simple groups were known. However, this depends on the unpublished work of Kleidman for the low-dimensional classical groups. Also, they are only “known” in the sense that the maximal subgroups of almost simple groups are well understood, and so it should be possible to produce a list of such groups. For applications we need the explicit list and no such list exists in the literature. Thus we produce such a list here, and we anticipate that the list will have many future applications, some can be seen in the subsequent chapters. One of the main results of this thesis is the following classification of finite primitive permutation groups with soluble point stabilizers.

Theorem 4.1.1. Let G ≤ Sym(Ω) be a ﬁnite primitive permutation group such that the point stabilizer H is soluble. Then one of the following holds:

d (i) G = Zp:H, and H ≤ GLd(p) is soluble and irreducible;

31 4.1. The main result

(ii) G is almost simple, and has a normal subgroup G0 which is minimal such that

H0 := H ∩ G0 is maximal in G0 and H = H0.o, where o = G/G0, and the pair

(G0,H0) is explicitly listed in TABLES I-VII, up to conjugacy. Moreover, with

the exceptions for which the numbers c of conjugacy classes of such subgroups H0

are listed in TABLE A, all of such subgroups H0 are conjugate.

m (iii) T ¡ G ≤ G1 o Sm, where G1 is an almost simple primitive group with socle T as in (ii), and the wreath product is in the product action.

TABLE A: Soluble maximal subgroups with more than one class

G0 H0 c conditions A6 S4 2 2 M12 3 :2S4 2 2 PSL3(3) 3 :2S4 2 2 PSU3(8) 3 :2A4 3 1+2 2 G2(3) (3+ × 3 ):2S4 2 PSL2(p)S4 2 p ≡ ±1 (mod 8) 2 PSL3(p) 3 :2A4 3 p ≡ 1 (mod 9) 2 PSU3(p) 3 :2A4 3 p ≡ 8 (mod 9)

In this thesis we mainly deal with the cases (ii) and (iii) in detail. The examples of almost simple groups with soluble maximal subgroups are presented in seven tables, where the maximal subgroups are listed up to conjugacy. Some of these lists are long, however, with a ﬁnite number of exceptions, these maximal subgroups are easily displayed. Recall that the socle of a group G is the subgroup of G generated by all its minimal normal subgroups, denoted by soc(G).

Corollary 4.1.2. Let G be an almost simple group which has a soluble maximal subgroup H. Then one of the following holds:

(i) G is a member of a collection of ﬁnitely many groups;

(ii) G = Ap or Sp, where p is a prime, with stabilizer Zp:Z p−1 or Zp:Zp−1, respectively; 2 + (iii) G is a classical group of Lie type of dimension at most 4, or soc(G) = PΩ8 (q);

(iv) soc(G) = PSLr(q) or PSUr(q) with r prime, H is a 1-dimensional group over Fqr ;

(v) G is an exceptional group of Lie type with soc(G) 6= E7(q).

Notice that, unlike the alternating groups and classical groups of Lie type which do not give many examples, exceptional groups of Lie type are indeed ‘exceptional’ in the sense that, except for E7(q), all of them have soluble maximal subgroups. The layout of this chapter is as follows. In Section 2 we prove Theorem 4.1.1 by reducing it to the classical groups of Lie type. In Section 3 we present all soluble

32 4.2. A reduction for the proof of Theorem 4.1.1

maximal Ci-subgroups of classical groups. In Sections 4–7 we determine the maximality

of the soluble maximal Ci-subgroups listed in Section 3, completing the proof of Theorem 4.1.1.

4.2 A reduction for the proof of Theorem 4.1.1

Here we prove Theorem 4.1.1 in the case where the group is not a classical group of Lie type.

Let G ≤ Sym(Ω) be primitive. Let H = Gω, where ω ∈ Ω. Suppose that H is soluble. Let N = soc(G) = T m, where T is simple and m is a positive integer. Then N

is transitive, and Nω ¡ Gω is soluble. ∼ m If N = Zp with p prime, then N is regular and Gω ≤ GLm(p) is soluble and irreducible, hence (i) of Theorem 1.1 holds.

Assume that T is non-abelian. If Nω = 1, then G is of twisted wreath product type, and by [24, Theorem 4.7B], Gω is isomorphic to a transitive subgroup of Sm whose point stabilizer has a composition factor isomorphic to T , which is impossible.

Hence Nω is a non-trivial soluble group. It follows that either m = 1 and G is almost simple, or m > 1 and G is of product action type. For the latter, G can be constructed from the almost simple type (see [24, Chapter 4]), and so the statement (iii) of Theorem 4.1.1 holds. Thus, to prove Theorem 1.1, we only need to determine soluble maximal subgroups of almost simple groups, up to conjugacy. A subgroup H < G is called local if H has a non-trivial soluble normal subgroup, and H is called p-local if it has a non-trivial normal p-subgroup with p prime. Obviously, soluble maximal subgroups are local subgroups. In the rest of this section, we prove Theorem 4.1.1 (ii) for the case where G is not a classical group of Lie type. First we ﬁx some important notation in this paper.

Notation. For an almost simple group G and a soluble maximal subgroup H of G,

deﬁne G0 with soc(G) ≤ G0 ¢ G to be minimal such that H0 := G0 ∩ H is a maximal ∼ subgroup of G0. Then G = G0.O, and H = H0.O, where O = G/G0 is a section of Out(T ).

Next we consider the cases where soc(G) is an alternating group, a sporadic simple group, or an exceptional simple group of Lie type in turn.

Lemma 4.2.1. Let G be An or Sn, with n ≥ 5. Then G0 ¢ G, and the pair (G0,H0)

is listed in TABLE I (see Section 4.8). In particular, either G0 = A6, which has two

conjugacy classes of subgroups S4, or there is only one conjugacy class of such subgroups

H0.

33 4.2. A reduction for the proof of Theorem 4.1.1

Proof. If n ≤ 13, by the Atlas [19], H is as in TABLE I, and either n = 6 and G0 = A6 has two conjugacy classes of subgroups S4, or all such subgroups H0 are conjugate. Thus, suppose n ≥ 14, and let G act naturally on ∆ = {1, 2, . . . , n}. The subgroup H is naturally divided into three classes.

First, assume that H is intransitive on ∆. Then H = (Sm ×Sn−m)∩G ≥ Am ×An−m.

Since n ≥ 14, at least one of Am and An−m is insoluble, and so H is insoluble, which is not possible.

Let H be transitive and imprimitive on ∆. Then n = md and H = (Sm o Sd) ∩ G. Suppose that n 6= 16. Then m ≥ 5 or d ≥ 5, and hence H is insoluble, which is a contradiction. Thus n = 16, and H = (S4 o S4) ∩ G. In this case H is the stabilizer of a 4 × 4 partition ∆ = ∆1 ∪ ∆2 ∪ ∆3 ∪ ∆4, where |∆i| = 4. Since G is transitive on all 4 × 4 partitions, H has a unique conjugacy class in G, as shown in TABLE I. Finally, suppose that H is primitive on ∆. Since n ≥ 14 and H is soluble, by the O’Nan-Scott theorem [24], we conclude that n = p with p prime, H is an aﬃne group ∼ and H ∩ Ap = Zp:Z p−1 . By [79], if p 6= 7, 11, 17, 23, then H ∩ Ap is maximal in Ap, as 2 in TABLE I. All subgroups of order p are conjugate in G, and H is the normalizer of a subgroup of order p. So H has a unique conjugacy class in G.

The next lemma shows that, except for M22,M24, and HS, every sporadic almost simple group has soluble maximal subgroups.

Lemma 4.2.2. Let G be a sporadic almost simple group with T = soc(G). Let H be a soluble maximal subgroup of G. Then G0 = T or T.2, G0 ¢G, and the pair (G0,H0) lies in TABLE II. Moreover, either G0 = M12, which has two conjugacy classes of subgroups 2 3 :2S4, or all such subgroups H0 are conjugate in G0.

Proof. By a recent paper [89], the list of the maximal 2-local subgroups of the Monster simple group in the Atlas [19] is complete. Thus all maximal local subgroups of sporadic almost simple groups are listed in [19]. Inspecting these subgroups, we conclude that the pairs (G0,H0) are as in TABLE II in section 10. Further, apart from the group M12, ∼ 2 which contains a soluble maximal subgroup H = 3 :2S4, with two conjugacy classes, in each case there is only one conjugacy class of such subgroups H0.

Lemma 4.2.3. Let G be an exceptional simple group of Lie type. Let H be a soluble maximal subgroup of G. Then the pair (G0,H0) lies in TABLE VII. Moreover, either 1+2 2 G0 = G2(3), which has two conjugacy classes of subgroups (3+ × 3 ):2S4, or all such subgroups H0 are conjugate in G0.

Proof. Assume that the characteristic of T = soc(G) is p. Since H is r-local for some prime r, H has a minimal normal r-subgroup E. Suppose ﬁrst that T = F4(q), Ei(q) 2 with i ∈ {6, 7, 8}, or E6(q). Then by [82], either

34 4.3. Subgroups of classical groups of Lie type

(a) r = p, and H is a maximal parabolic subgroup, or

r 6= p, and one of the following three statements holds:

(b) H is of maximal rank, classiﬁed in [82];

(d) H = CT (α) for some outer automorphism α of T of prime order, and H is insoluble (see [12, Proposition 2.7]).

As H is soluble, H does not satisfy item (d). If H is a maximal parabolic subgroup, as in part (a), then H is an extension of an r-group by the Chevalley group determined by a maximal subdiagram of the Dynkin diagram of G. It follows that H is insoluble, which is not possible. Suppose that H is a subgroup of maximal rank, as in part (b). Then inspecting the

Table 5.1 and Table 5.2 of [82], we conclude that the pairs (G0,H0) are as in TABLE VII. Suppose that H is not of maximal rank, as in part (c). Then inspecting [12], Table 1, 0 ∼ 3 there is only one soluble case: G0 = Ree(3) = PSL2(8), and H0 = 2 :7, which has been included in TABLE III. For the remaining cases, all maximal subgroups of G are completely known: see 2 3 [113] for T = Sz(q), [85] for T = F4(q), [61] for T = Ree(q), [60] for T = D4(q) and

[20, 61] for T = G2(q). Inspecting these lists, we obtain that the pairs (G0,H0) lie in ∼ 1+2 TABLE VII. Furthermore, if G0 = G2(3), H0 = (3+ ):2S4, then H0 has two classes in G0. In the other cases, subgroups isomorphic to H0 form a unique conjugacy class.

Thus, to complete the proof of Theorem 4.1.1, there remains only the case of the classical groups of Lie type. Since the work of Kleidman on the subgroups of classical groups of dimension at most 12 has never been published, R. A. Wilson recently suggests to attack the following problem: “Re-do Kleidman’s work on the maximal subgroups of classical groups in dimension up to 12 (and possibly more), needs great care and accuracy” (on his homepage, Problem 4). Our task here is to give an explicit list of almost simple groups which have a soluble maximal subgroup. For classical groups the analysis will depend mainly on Kleidman-Liebeck’s monograph [62], together with the Atlas [19], and will be accomplished in Propositions 4.4.1, 4.5.1, 4.6.1 and 4.7.2.

4.3 Subgroups of classical groups of Lie type

Let G be an almost simple group, with T = soc(G) a classical group of Lie type. As- chbacher [1] classiﬁed maximal subgroups of G into eight classes of geometric subgroups

35 4.3. Subgroups of classical groups of Lie type and a class of almost quasisimple subgroups. In the monograph [62], Kleidman and Liebeck gave detailed description about Aschbacher’s classes. Here we will adopt the deﬁnition and description given in Chapter 4 of [62] to identify maximal soluble subgroups of G. We remark that there is a slight diﬀerence between the two.

4.3.1 Preliminaries

Let M be a maximal soluble subgroup of G. Then M lies in one of the eight classes of geometric subgroups of G, called Ci, for 1 ≤ i ≤ 8 (in the sense of Kleidman-Liebeck).

For convenience, by a Ci-subgroup M we mean that M ∈ Ci. The following Aschbacher Theorem is the key tool for proving Theorem 1.1.

Theorem 4.3.1. [1, Aschbacher 1985] Let G be a ﬁnite almost simple classical group of Lie type with socle T , and let H be a maximal subgroup of G. Then one of the following holds:

(i) H ∩ T is a maximal Ci-subgroup of T for some i ≤ 8;

+ f (ii) T = PΩ8 (q) and G contains a triality automorphism, or T = PSp4(2 ) and G contains a graph automorphism;

(iii) H ∈ C9 is almost quasisimple, and is absolutely irreducible on V .

Moreover, each subgroup K < T is a Ci-subgroup for some i ≤ 9.

f + We remark that the maximal subgroups of G with soc(G) = PSp4(2 ) or PΩ8 (q) are known, see section 4 and section 6 for detail. Note also that there exist subgroups which may be Ci-subgroups and also Cj-subgroups for i 6= j. Furthermore, the maximal

Ci-subgroups of T are not necessarily maximal subgroups, see Example 4.3.2 below.

r Example 4.3.2. Let T = PSL2(2 ) with r ≥ 3 prime. Then a maximal C5-subgroup ∼ M = PSL2(2) = S3 is not maximal in T because it is contained in a maximal Cj-subgroup ∼ r ∼ r K, where j = 2 and K = D2(2r−1) if 3 | (2 − 1), or j = 3 and K = D2(2r+1) if 3 | (2 + 1).

In this example, if r = 6, then the maximal C2-subgroup K = D2.63 has a proper ∼ subgroup D2.21 which properly contains the maximal C2-subgroup M = S3. This shows that a maximal Ci-subgroup may be a proper subgroup of a non-maximal Cj-subgroup.

In this section, we will list all soluble maximal Ci-subgroups of classical simple groups of Lie type, and then determine the maximality for each of them later. We ﬁrst make the following observations regarding small classical groups and the isomorphisms between diﬀerent types of classical groups. Except for 1-dimensional classical groups, we have the following soluble classical groups:

36 4.3. Subgroups of classical groups of Lie type

∼ ∼ ∼ ∼ ∼ (1) Sp2(2) = SU2(2) = GL2(2) = SL2(2) = PSL2(2) = S3;

∼ ∼ ∼ ∼ (2) Sp2(3) = SU2(3) = SL2(3) = Q8.3 = 2.A4;

∼ ∼ ∼ ∼ Ω3(3) = PSp2(3) = PSU2(3) = PSL2(3) = A4;

∼ ∼ ∼ GL2(3) = Q8:S3 = 2.S4; PGL2(3) = S4.

+ ∼ + ∼ + ∼ (3)O 2 (q) = D2(q−1), SO2 (q) = Zq−1.Z(2,q), and Ω2 (q) = Z(q−1)/(2,q−1);

− ∼ − ∼ − ∼ O2 (q) = D2(q+1), SO2 (q) = Zq+1.Z(2,q), and Ω2 (q) = Z(q+1)/(2,q−1);

∼ 2 (4) PSU3(2) = 3 :Q8.

+ ∼ ∼ + ∼ ∼ (5)Ω 4 (2) = SL2(2) × SL2(2) = S3 × S3;Ω4 (3) = SL2(3) ◦ SL2(3) = 2.(A4 × A4).

Some of the isomorphism relations between families of classical groups are as follows:

∼ ∼ ∼ (1)SL 2(q) = Sp2(q) = SU2(q), and PSL2(q) = Ω3(q) for q odd;

+ ∼ ∼ − ∼ 2 (2)Ω 4 (q) = SL2(q) ◦ SL2(q) = (2, q − 1).(PSL2(q) × PSL2(q)); Ω4 (q) = PSL2(q );

∼ ∼ (3) For q odd, PSp4(q) = Ω5(q); for q even, Ω2m+1(q) = Sp2m(q) for m ≥ 1;

+ ∼ − ∼ (4)PΩ 6 (q) = PSL4(q); PΩ6 (q) = PSU4(q).

4.3.2 Soluble maximal Ci-subgroups

Here we will produce a list of soluble maximal Ci-subgroups, for 1 ≤ i ≤ 8. Let T be an n-dimensional classical simple group over GF(q) with q = pf and p prime. Due to the above generic isomorphisms, as usual, we make the assumptions: n ≥ 3 for unitary groups; n ≥ 4 for symplectic groups; n ≥ 7 for orthogonal groups.

Let M be a soluble maximal Ci-subgroup of T , and let c be the number of the conjugacy classes of M in T .

Lemma 4.3.3. Let M ∈ C1. Then T , M, and c are listed in the following table.

37 4.3. Subgroups of classical groups of Lie type

T M c conditions f PSL2(q) Zp :Z(q−1)/k 1 k = (2, q − 1) 2 PSL3(3) 3 :2S4 2

PSU3(3) 4.S4 1 3 2 PSL3(q) ([q ]:Zq−1)/Z(3,q−1) 1 3 PSU3(q) ([q ].Zq2−1)/Z(3,q+1) 1 q ≥ 3 4 PSL4(2) 2 :(S3 × S3), 1 5 [2 ]:S3 1 4 PSL4(3) 3 :2(A4 × A4).2, 1 5 [3 ]:2S4 1

PSU4(2) 2(A4 × A4).2, 1 3 [3 ]:S4 1 5 PSU4(3) [3 ].2S4 1

PSp4(2) S4 × 2 2 8 2 PSL5(q) [q ]:GL2(q) 1 q = 2 or 3 7 3 PSU5(2) [2 ]:[3 ]:2A4, 1 3 S3.[3 ]:2A4 1 12 3 (q−1)2 PSL6(q) [q ]:SL2(q) .[ (q−1,6) ] 1 q = 2 or 3 7 PSp6(q) [q ]:(GL2(q) ◦ Sp2(q)) 1 q = 2 or 3 7 Ω7(3) [3 ].(2A4 × A4).2, 1 2 S4 × 2.A4.2 1 + 9 3 PΩ8 (2) [2 ]:S3 1 + 9 3 PΩ8 (3) [3 ]:2.A4.2 1

Proof. Maximal C1-subgroups are described in [62, Section 4.1]. Since M is soluble, it follows that n ≤ 8. Inspecting the candidates given in [62, Section 4.1], we conclude ∼ ∼ 2 that either T = PSL3(3) has two conjugacy classes of subgroups M = P1 = 3 :2S4, or the soluble maximal C1-subgroups are conjugate, and one of the following cases appears. ∼ f (i) T = PSL2(q), and M = Zp :Z(q−1)/k, where k = (2, q − 1). ∼ (ii) n = 3, and one of the following cases occurs: T = PSU3(3), and M = 4.S4; ∼ ∼ 3 2 ∼ T = PSL3(q), and M = P1,2 = ([q ].Zq−1)/Z(3,q−1); T = PSU3(q), and M = 3 ([q ].Zq2−1)/Z(3,q+1).

∼ 4 5 (iii) n = 4, and T = PSL4(2), and M = 2 :(S3 × S3) or [2 ]:S3; or T = PSL4(3), ∼ 4 5 ∼ and M = 3 :2(A4 × A4).2 or [3 ]:2S4; or T = PSU4(2) = PSp4(3), and M = 3 ∼ 5 2(A4 × A4).2 or [3 ]:S4; or T = PSU4(3), and M = [3 ].2S4; or T = PSp4(2), and ∼ M = S4 × 2.

∼ 8 2 (iv) n = 5, and either T = PSL5(q) with q = 2 or 3, and M = [q ]:GL2(q) , or ∼ 7 3 3 T = PSU5(2), and M = [2 ]:[3 ]:2A4 or S3.[3 ]:2A4.

38 4.3. Subgroups of classical groups of Lie type

∼ 12 3 (q−1)2 (v) n = 6, and either T = PSL6(q), with q = 2 or 3, and M = [q ]:SL2(q) .[ (q−1,6) ], ∼ 7 or T = PSp6(q) with q = 2 or 3, and M = [q ] : (GL2(q) ◦ Sp2(q)).

∼ 7 2 (vi) T = Ω7(3), and M = [3 ].(2A4 × A4).2 or S4 × 2.A4.2.

+ ∼ 9 3 + ∼ (vii) n = 8, and either T = PΩ8 (2) and M = [2 ]:S3, or T = PΩ8 (3) and M = 9 3 [3 ]:2.A4.2.

All the candidates T and M are listed in the above table, and the proof is completed .

Lemma 4.3.4. Let M ∈ C2. Then T , M, and c are listed in the following table.

T M c conditions

PSL2(q) D2(q−1)/(2,q−1) 1 q ≥ 4 2 PSL3(q) [(q − 1) /(3, q − 1)].S3 1 2 PSU3(q) [(q + 1) /(3, q + 1)].S3 1 q ≥ 3 3 PSL4(q) [(q − 1) /(4, q − 1)].S4 1 3 PSU4(q) [(q + 1) /(4, q + 1)].S4 1 t PSL2t(2) PSL2(2) .St 1 2 ≤ t ≤ 4 t−1 t t−1 PSL2t(3) [2 ].PSL2(3) .[2 ].St 1 2 ≤ t ≤ 4 (q+1)t−1(q+1,2) t t−1 PSU2t(q) [ (q+1,2t) ].PSL2(q) .[(q + 1, 2) ].St 1 2 ≤ t ≤ 4, q = 2 or 3 t−1 PSp2t(q) (2, q − 1) .(PSp2(q) o St) 1 2 ≤ t ≤ 4, q = 2 or 3 t−1 t t−1 PSU3t(2) [3 ].PSU3(2) .[3 ].St 1 2 ≤ t ≤ 4

PSp4(3) PGL2(3).2 1 + + 2 PΩ8 (2) Ω4 (2) .2.2 1 + + 2 2 PΩ8 (3) 2.PΩ4 (3) .2 .2 1 + ± 4 3 PΩ8 (q) Ω2 (q) .2 .S4 1 q even + 3 + 4 6 PΩ8 (q) 2 .PΩ2 (q) .2 .S4 1 (q − 1)/2 even 2 − 4 3 2 .Ω2 (q) .2 .S4 1 (q − 1)/2 even 3 − 4 6 2 .PΩ2 (q) .2 .S4 1 (q − 1)/2 odd 2 + 4 3 2 .Ω2 (q) .2 .S4 1 (q − 1)/2 odd 2 3 2 Ω9(3) (2 × Ω3(3) .2 ).S3 1 + + 3 2 PΩ12(2) Ω4 (2) .2 .S3 1 + 2 4 3 PΩ12(3) 2 × Ω3(3) .2 .S4 2 2 + 3 4 2 .(PΩ4 (3)) .2 .S3 1 + + 4 3 PΩ16(2) Ω4 (2) .2 .S4 1 + 3 + 4 6 PΩ16(3) 2 .(PΩ4 (3)) .2 .S4 1

Proof. By deﬁnition, C2-subgroups M are stabilizers of subspace decompositions V =

V1 ⊕ V2 ⊕ ... ⊕ Vt, where dim(Vi)=m, and n = mt. The structure of M is described in [62, Section 4.2]. Inspecting these candidates, we have the following statements.

39 4.3. Subgroups of classical groups of Lie type

2 The subgroups in [62, Prop 4.2.4] are not soluble, because PSLn/2(q ) is insoluble. The candidates in [62, Prop 4.2.5] are soluble only in the case where n = 4 and ∼ q = 2 or 3. Since PSp4(2) = S6, we only have one case which needs considering, which ∼ is T = PSp4(3), and M = PGL2(3).2.

The subgroups in [62, Prop 4.2.7] are insoluble, since it involves PSLn/2(q) with n ≥ 8. For the candidates in [62, Prop 4.2.9], the solubility of M implies that n = mt such that 2 ≤ t ≤ 4 and either m = 1, or m = 2 and q = 2 or 3, or m = 3, q = 2, and

T = PSUn(q). In this case, either

t−1 t t−1 (i) T = PSLn(q), and M = [(q −1) (q −1, m)/(q −1, n)].PSLm(q) .[(q −1, m) ].St, or

t−1 t t−1 (ii) T = PSUn(q), and M = [(q+1) (q+1, m)/(q+1, n)].PSUm(q) .[(q+1, m) ].St.

t−1 The candidates of [62, Prop 4.2.10] lead to T = PSpn(q) and M = (2, q−1) .(PSp2(q)o

St), where n = 2t such that 2 ≤ t ≤ 4 and q = 2 or 3. The other candidates are given in Propositions 4.2.11 and 4.2.14-4.2.16 of [62], for which T is an orthogonal group, so n ≥ 7. The subgroups given in Propositions 4.2.15- + 4.2.16 of [62] are insoluble. If M satisﬁes Proposition 4.2.11 or 4.2.14, then T = PΩn (q), where n = mt such that m ≥ 2 and t = 2, 3 or 4. Thus, we have one of the following cases.

+ ± 4 3 (1) m = 2, t = 4 and T = PΩ8 (q), satisfying [62, Prop 4.2.11]. Here M = Ω2 (q) .2 .S4 3 + 4 6 2 − 4 3 for q even; or M = 2 .PΩ2 (q) .2 .S4 or 2 .Ω2 (q) .2 .S4 for (q − 1)/2 even; or 3 − 4 6 2 + 4 3 M = 2 .PΩ2 (q) .2 .S4 or 2 .Ω2 (q) .2 .S4 for (q − 1)/2 odd.

∼ 2 3 2 + (2) m = 3, and either T = Ω9(3), and M = (2 × Ω3(3) .2 ).S3, or T = PΩ12(3), and ∼ 2 4 3 M = 2 × Ω3(3) .2 .S4;

+ ∼ (3) m = 4, and one of the following six cases occurs: T = PΩ8 (2), and M = + 2 + ∼ + 2 2 + ∼ Ω4 (2) .2.2; T = PΩ8 (3), and M = 2.PΩ4 (3) .2 .2; T = PΩ12(2), and M = + 3 2 + ∼ 2 + 3 4 + Ω4 (2) .2 .S3; T = PΩ12(3), and M = 2 .(PΩ4 (3)) .2 .S3; T = PΩ16(2), and ∼ + 4 3 + ∼ 3 + 4 6 M = Ω4 (2) .2 .S4; T = PΩ16(3), and M = 2 .(PΩ4 (3)) .2 .S4.

+ ∼ 2 4 3 Finally, we point out that either c = 1, or T = PΩ12(3), and M = 2 × Ω3(3) .2 .S4, and c = 2, as shown in the table.

Next we consider the soluble maximal C3-subgroups.

Lemma 4.3.5. If M ∈ C3, then (T, M, c) lies in the following table, where r is a prime.

40 4.3. Subgroups of classical groups of Lie type

T M c PSLr(q) Z(qr−1)/(q−1)(r,q−1):Zr 1 PSUr(q) Z(qr+1)/(q+1)(r,q+1):Zr 1 PSp4(3) 2PGU2(3).2 1

Proof. Maximal C3-subgroups are described in a series of lemmas of [62, Section 4.3].

For the candidates in [62, Prop 4.3.6], the solubility of M implies that T = PSLr(q) ∼ with r prime, M = Z(qr−1)/((q−1)(r,q−1)):Zr, or T = PSUr(q) with r prime, M =

Z(qr+1)/((q+1)(r,q+1)):Zr. In both cases, c = 1. ∼ The candidate in [62, Prop 4.3.7] gives rise to T = PSp4(3), M = 2.PGU2(3).2, and c = 1. r For the candidates in [62, Prop 4.3.10], M = PSpm(q ).r is insoluble, where r is prime. Similarly, the candidates in Propositions 4.3.14, 4.3.16-4.3.18, 4.3.20 of [62] are insoluble.

Lemma 4.3.6. If M ∈ C4, then T , M, and c are listed in the following table.

T M c

PSU6(2) PSU2(2) × PSU3(2) 1

PSp6(3) PSp2(3) × SO3(3) 1 + PSp8(3) (PSp2(3) × PO4 (3)).2 1 + + PΩ12(3) PΩ4 (3) × SO3(3) 1

Proof. A C4-subgroup M is the stabilizer of a tensor decomposition V = U ⊗ W , where dim(U) 6= dim(W )[62, Chapter 4]. Further, if dim(U) < dim(W ), then dim(U) ≥ 2, dim(W ) ≥ 3, and so n ≥ 6. We point out that in almost all cases, a maximal C4-subgroup of T has an insoluble composition factor S, as indicated in the following discussion. ∼ If T = PSLn(q), since n ≥ 6, by [62, Prop 4.4.10], S = PSLm(q) with m ≥ 3.

Suppose that T = PSUn(q). Then by [62, Prop 4.4.10], for (n, q) = (6, 2), the ∼ subgroup M = PSU2(2) × PSU3(2) is soluble, and c = 1; for n ≥ 6 and (n, q) 6= (6, 2), ∼ we have that S = PSUm(q) with m ≥ 3 and (m, q) 6= (3, 2).

Suppose that T = PSpn(q). By [62, Prop 4.4.11], there are only two soluble cases: ∼ ∼ (n, q) = (6, 3) with M = PSp2(3) × SO3(3), and c = 1; or (n, q) = (8, 3) with M = + ∼ (PSp2(3) × PO4 (3)).2, and c = 1. For the other cases, S = PSpm(q) with m ≥ 4 is an insoluble composition factor. ∼ If T = Ωn(q) with nq odd, then n ≥ 15, and by [62, Prop 4.4.18], S = Ωm(q) with m ≥ 5. − ∼ − If T = PΩn (q) with n even, then by [62, Prop 4.4.17], S = PΩm(q) with m ≥ 4.

41 4.3. Subgroups of classical groups of Lie type

+ Suppose that T = PΩn (q) with n even. Then by [62, Prop 4.4.14∼17], the only ∼ + soluble case is (n, q) = (12, 3), with M = PΩ4 (3) × SO3(3), and c = 1. For if (n, q) 6= ∼ ∼ (12, 3), then either S = PΩm(q) with m ≥ 4 and q ≥ 5, or S = PSpm(q) with m ≥ 4.

Lemma 4.3.7. If M ∈ C5, then (T, M, c) lies in the following table, where r is a prime.

T M c conditions r PSL2(2 ) PSL2(2) 1 r ≥ 2 PSL2(9) PGL2(3) 2 r PSL2(3 ) PSL2(3) 1 r ≥ 3 r PSU3(2 ) PSU3(2).(3, r) (3, r) r ≥ 3 PSU3(3) PSO3(3) 1 + PSU4(3) PSO4 (3).2 2

Proof. Maximal C5-subgroups are given in a series of lemmas in [62, Section 4.5]. Among r the candidates in [62, Prop 4.5.3], the soluble ones are the following: T = PSL2(2 ), r M = PSL2(2), r ≥ 2, and c = 1; T = PSL2(9), M = PGL2(3), and c = 2; T = PSL2(3 ), r M = PSL2(3), and c = 1; T = PSU3(2 ), M = PSU3(2).(3, r), r ≥ 3 prime, and c = (3, r). 1/r The candidates in [62, Prop 4.5.4] are insoluble since they are isomorphic to PSpn(q ).c.

For the candidates in [62, Prop 4.5.5], we have T = PSU3(3), M = PSO3(3), and + c = 1, together with T = PSU4(3), M = PSO4 (3).2, and c = 2. All the candidates in [62, Prop 4.5.8, 4.5.10] are insoluble.

Lemma 4.3.8. If M ∈ C6, then (T, M, c) lies in the following table, where p is a prime.

T M c conditions PSL2(p)A4 1 p ≡ ±3 (mod 8) S4 2 p ≡ ±1 (mod 8) 2 PSL3(p) 3 :Q8 1 p 6≡ 1 (mod 9), 3|(p − 1) 2 3 :Q8.3 3 p ≡ 1 (mod 9) 2 PSU3(p) 3 :Q8 1 p 6≡ 8 (mod 9), 3 | (p − 2) 2 3 :Q8.3 3 p ≡ 8 (mod 9)

Proof. Maximal C6-subgroups are determined in Propositions 4.6.5-4.6.9 of [62, Section 4.6]. The candidates in Propositions 4.6.6, 4.6.8 and 4.6.9 are all insoluble. ∼ For the candidates in [62, Prop 4.6.7], we have T = PSL2(q), M = A4.a, and c = a, where a ≤ 2, a = 1 if q ≡ ±3 (mod 8), and a = 2 if q = p ≡ ±1 (mod 8). For the candidates in [62, Prop 4.6.5], we have two cases:

∼ 2 (1) T = PSL3(q), M = 3 :Q8.b, q = p ≡ 1 (mod 3), and c = b, where b | 3, b = 3 if and only if q = p ≡ 1 (mod 9); and

42 4.3. Subgroups of classical groups of Lie type

∼ 2 (2) T = PSU3(q), M = 3 :Q8.e, q = p ≡ 2 (mod 3), c = e, where e | 3, e = 3 if and only if q = p ≡ 8 (mod 9).

Lemma 4.3.9. All maximal C7-subgroups are insoluble.

Proof. By [62, Lemma 4.7.1], all maximal C7-subgroups are non-local and so are insoluble.

Lemma 4.3.10. If M ∈ C8, then (T, M, c) lies in the following table, where r is a prime.

T M c condition ± PSL2(q) PSO2 (q).2 1 q odd PSL3(3) PSO3(3) 1 PSL3(4) PSU3(2) 1 + PSL4(3) PSO4 (3).2 1 + PSp4(2) Ω4 (2).2 1

Proof. A maximal C8-subgroup is a classical group of dimension n, and is determined in [62, Section 4.8]. We inspect these candidates.

Each candidate in [62, Prop 4.8.3] has an insoluble composition factor PSpn(q), where n ≥ 4.

For the candidates in [62, Prop 4.8.4], we have that c = 1, and T = PSL2(q), M = ± + PSO2 (q).2 with q odd; T = PSL3(3), M = PSO3(3); or T = PSL4(3), M = PSO4 (3).2.

The only soluble candidate in [62, Prop 4.8.5] is T = PSL3(4), M = PSU3(2), and c = 1. + For the candidates in [62, Prop 4.8.6], we have T = PSp4(2), M = Ω4 (2).2, and c = 1.

4.3.3 The maximality

Our remaining task is to determine the maximality of the soluble subgroups M given in Lemmas 4.3.3-4.3.10.

We remark that for n ≥ 13, the maximality of maximal Ci-subgroups is determined by Kleidman and Liebeck in [62]. Thus, for n ≥ 13, we can distill our result directly from [62]. By Lemmas 4.3.3-4.3.10, the only groups T of dimension n ≥ 13 which have + + soluble maximal Ci-subgroups are PΩ16(2), or PΩ16(3), or PSLr(q) or PSUr(q) with r prime and M ∈ C3. The maximality of M ∈ C3 in PSLr(q) or PSUr(q) can be veriﬁed in + + a uniform way considered later, and we quote [62] for the groups PΩ16(2) and PΩ16(3). On the other hand, there is an extensive literature concerning the maximal subgroups of the low-dimensional classical groups. We will review the literature for each class of classical groups, and refer to the references with which we are familiar.

43 4.3. Subgroups of classical groups of Lie type

If M ∈ Ci for 1 ≤ i ≤ 8, an overgroup K of M may be a geometric subgroup, that is, a

Cj-subgroup where 1 ≤ j ≤ 8, or a subgroup which is not a geometric subgroup, called a

C9-subgroup for convenience. A C9-subgroup is absolutely irreducible. For convenience, a subgroup X of T is said to be (absolutely) irreducible if its preimage Xˆ in GLn(q) is (absolutely) irreducible on the n-dimensional space V .

Following [62], each class Ci breaks up into several subclasses which are called types.

For example, the C2 class of the unitary group PSU6(q) is split into four types: type 2 GUm(q) o St, where m ∈ {1, 2, 3} and mt = 6, and type GL3(q ).2. Here we have a simple property presented below, which is important for deciding whether a maximal Ci-subgroup is maximal or not.

Lemma 4.3.11. Let T be a classical simple group of Lie type, and let M ∈ Ci for i ≤ 8. If M is not a maximal subgroup of T , then either

(i) M < K for some maximal Cj-subgroup K of T , and M, K are not of the same type, or

(ii) each overgroup of M in T is almost quasisimple and lies in C9.

Proof. Let L be an overgroup of M which is not almost quasisimple. By the Aschbacher

Theorem, there exists a maximal Cj-subgroup K that contains L. Then, by [62, Theorem 3.1.1], M and K are not of the same type, as in part (i).

A maximal Ci-subgroup is not necessarily a maximal subgroup. For example, G =

PGL2(7) has a maximal subgroup H = D16, however, H ∩ T = D8 is not maximal in

T , as it is contained in K = S4 < T . Such a maximal subgroup H of G is said to be a novelty, that is, a maximal subgroup H of G is called a novelty if H ∩ T is not maximal in T .

Example 4.3.12. Let T = PSLn(q) with n ≥ 3, and let

(q − 1)2 M = P = [q2n−3]:[ ].PSL (q).(n − 2, q − 1), 1,n−1 (n, q − 1) n−2 which is the stabilizer of a pair (U, W ) where U < W , dim(U) = 1 and dim(W ) = n − 1.

Then P1,n−1 is contained in the parabolic subgroups P1 and Pn−1, and P1, Pn−1 are not conjugate in T . Let G = T.O.hσi, where O ≤ PΓLn(q)/PSLn(q), and σ is the σ transpose-inverse map. Then P1 = Pn−1, and thus NG(M) = M.O.hσi is a novelty maximal subgroup of G. In particular, if n = 3, or (n, q) = (4, 2) or (4, 3), then M is soluble.

The next lemma provides a method to construct and study novelty maximal subgroups.

44 4.4. Subgroups of linear groups

Lemma 4.3.13. Let G be a classical almost simple group with socle T . Let M be a

maximal Ci-subgroup of T with i ≤ 8 which is not a maximal subgroup of T . Assume

that NG(M) is a maximal subgroup of G. Then for each K with M < K < T , either σ M has at least two conjugacy classes in K, or there exists σ ∈ NG(M) such that K,K are not conjugate in T .

Proof. Since NG(M) is a maximal subgroup of G that contains M, we have NG(M) = M.O, where O = G/T . Suppose that for some K with M < K < T , M has a unique σ k conjugacy class in K. Then for each σ1 ∈ G, there exists k ∈ K such that M = M 1 . σ k Assume that for every σ1 ∈ G, the subgroups K,K 1 are conjugate in T . Then there

t σ1k −1 ∼ exists t ∈ T such that K = K . Thus σ1kt ∈ NG(K). It follows that T NG(K)/T =

G/T , and hence M.O < K.O = NG(K) < G, which is a contradiction. Thus for some σ σ1 ∈ G, K,K are not conjugate in T , where σ = σ1k ∈ NG(M).

For positive integers q and n, a prime divisor of qn − 1 is called a primitive prime divisor if it does not divide qi − 1 for all i < n. The well-known result of Zsigmondy will be used frequently.

Theorem 4.3.14. For any positive integers q and n, either qn −1 has a primitive prime divisor, or (n, q) = (6, 2) or (2, 2b − 1), where b ≥ 2 is an integer.

Lemma 4.3.15. For any positive integers q, n with q ≥ 2 and n ≥ 3, assume that (n, q) 6= (6, 2) or (2, 2b −1), and p is a primitive prime divisor of qn −1. Then n|(p−1).

Proof. Since p|(qn − 1), we have (p, q) = 1. Thus by Euler’s Theorem, qϕ(p) = qp−1 ≡ 1(mod p). Since p is a primitive prime divisor of qn − 1, we obtain p − 1 ≥ n. Assume that p − 1 = an + b, where 0 ≤ b < n, we show that b = 0. Suppose that 0 < b < n. Then qp−1 − 1 = (qan+b − qb) + (qb − 1), and so p|(qb − 1), a contradiction. Thus b = 0 and n|(p − 1).

We end this section with some notation. For a group G and a prime divisor p of |G|, let Gp be a Sylow p-subgroup of G. Then |G|p, which is the highest power of p dividing

|G|, is equal to the order |Gp|.

4.4 Subgroups of linear groups

f Let T = PSLn(q), where q = p with p prime, and let M be a soluble maximal Ci- subgroup of T , with 1 ≤ i ≤ 8. Then the pair (T,M) is given in Lemmas 4.3.3-4.3.10. For small n, say for n ≤ 5, the maximality of M has been completely or partially determined for a long time. For n = 2, the subgroup structure of T was determined by Moore [92] and Wiman [132], based on which Dickson gave a treatment in his book [23].

45 4.4. Subgroups of linear groups

Complete information on the maximal subgroups of G, where T ≤ G ≤ Aut(T ), can be found in [113] or [42]. For n = 3, the maximal subgroups of T were determined by Mitchell [90] and Hartley [48] (see King [58] for references). For n = 4, when q is even, the maximal subgroups of PSL4(q) were determined by Mwene [93]; and when q is odd, more or less the maximal subgroups have been determined by Mwene [94], Zalesski and Suprunenko [134] (also see [57]). For n = 5, some maximal subgroups of T have been known [57]. Also for some i, the maximality of the maximal Ci-subgroups of T has been known (see [58] for references). The maximality of M and the possible occurrence of novelties are described as follows.

Proposition 4.4.1. Either M is a maximal subgroup of T , and (T,M) = (G0,H0) lies in TABLE III, or M < K < T is such that the triple (T,M,K) lies in TABLE III0, 0 and moreover, G0 in TABLE III is such that T < G0 ≤ Aut(T ), G0 contains a novelty

H0, and H0 ∩ T = M. In particular, with the exceptions listed in TABLE A, all such subgroups H0 are conjugate.

Proof. By Lemmas 4.3.3-4.3.10, the candidates for T consist of two inﬁnite families:

PSLr(q) with r prime and PSL4(q), and a few small groups which we analyse separately. As mentioned at the beginning of this section, maximal subgroups of G with T = 0 PSL2(q) are all known and are listed in TABLE III and TABLE III , in particular, for q = p ≡ ±1 (mod 8), T has two classes of maximal subgroups S4. Thus, we assume that n ≥ 3. Assume that M is not maximal in T , and let K be such that M < K < T .

∼ Case 1. Let T = PSLr(q) with r ≥ 3 prime, and M = Z(qr−1)/(q−1)(r,q−1):Zr.

Then M ∈ C3 is irreducible, and so is K. Hence K is not a C1-subgroup. By Lemma

4.3.11, K/∈ C3, and since r is a prime, T has neither a C4- nor a C7-subgroup. If

K ∈ C2 ∪ C5 ∪ C6, then it is easily shown that |M| does not divide |K|, which is not ∼ 2 ∼ possible. Suppose that K ∈ C8. Then K = PSUr(q0) with q = q0, or K = PSOr(q). ∼ Since the order |PSOr(q)| is not divisible by |M|, we have K = PSUr(q0). r Suppose that p0 is a primitive prime divisor of q0 − 1. As |M| | |K| and |K| = 1 qr(r−1)/2(q2 − 1)(q3 + 1) ... (qr + 1), we have that p divides qr(r−1)/2(q2 − 1)(q3 + (r,q0+1) 0 0 0 0 0 0 0 0 r−1 m 1) ... (q0 − 1). Thus p0 divides q0 + 1 for some 3 ≤ m ≤ r − 2, and so p0 divides r m m r−m t (q0 − 1) + (q0 + 1) = q0 (q0 + 1). It follows that p0 divides (q0 + 1), where t = 2t min{m, r − m}. Thus p0 divides q0 − 1, which is a contradiction as 2t < r. Therefore, r q0 − 1 has no primitive prime divisor.

46 4.4. Subgroups of linear groups

TABLE III0

( Pm and Pm,n−m are parabolic subgroups of PSLn(q), and σ is the transpose-inverse)

TMKG0 H0 conditions PSL2(7) D6 S4 PGL2(7) D12 D8 S4 PGL2(7) D16 PSL2(9) D8 S4 PGL2(9) D16 M10 Z8:Z2 5 PΓL2(9) [2 ] D10 A5 PGL2(9) D20 M10 Z5:Z4 PΓL2(9) Z10:Z4 PSL2(11) D10 A5 PGL2(11) D20 r PSL2(2 ) PSL2(2) D2(2r+1) − − r > 2 PSL2(p)A4 A5 PGL2(p)S4 p ≡ ±11, ±19 (40) PSL3(4) 7:3 PSL2(7) PGL3(4) 7:3 × 3 2 3.S3 3 :Q8 − − PSL3(q) P1,2 P1,P2 T.hσi M.hσi σ graph auto PSL3(q) P1,2 P1,P2 T.hτσi M.hτσi σ graph auto, τ ﬁeld auto PSL4(2) S4 PSp4(2) − − 2 S3.2 PSp4(2) − − 4 3 4 2 :A4 2 :PSL3(2) T.2 2 :S4 2 PSL4(3) 2 .S4 (4 × A6):2 − − 2 2.A4.2.2 PSU4(2).2 PGL4(3) 2(S4 × S4).hδi δ diagonal auto 2 2.A4.2.2 PSU4(2).2 T.hσδi 2(S4 × S4).hσδi σ graph, δ diagonal auto 3 PSL4(4) 3 .S4 PSU4(2) − − 2 4 3 PSL4(5) 4 .S4 2 :A6 PGL4(5) 4 :S4 2 4 3 PSL4(5) 4 .S4 2 :A6 PSL4(5).hδσi 4 :S4 σ graph, δ diagonal auto PSL4(p) P1,3 P1,P3 T.hσi M.hσi p = 2, 3 PSL4(3) P1,3 P1,P3 T.hσδi M.hσδi σ graph, δ diagonal auto PSLd(p) P2,d−2 P2,Pd−2 T.hσi M.hσi d = 5, 6; p = 2, 3 PSL6(3) P2,4 P2,P4 T.hσδi M.hσδi σ graph, δ diagonal auto 3 PSL6(2) S3.S3 PSp6(2) − − 4 PSL8(2) S3.S4 PSp8(2) − −

r 6 Since r ≥ 3 is a prime, by Lemma 4.3.14, we have that q0 = 2 , r = 3 and q0 = 4, 2 ∼ ∼ that is, T = PSL3(4 ) and K = PSU3(4). Thus, M = Z(162+16+1)/3:Z3. However, a metacyclic subgroup of K that has the largest order is Z42−4+1:Z3, which contradicts the preceding sentence.

Thus, K is not a Ci-subgroup for 1 ≤ i ≤ 8, and by Lemma 4.3.11, each overgroup of M is almost quasisimple of type C9. Choose K to be a minimal one. Let K = K/Z(K), and M = MZ(K)/Z(K). Then K is almost simple and has a soluble maximal subgroup M. Thus, (soc(K), M) is a pair of groups appearing in Lemmas 4.3.3-4.3.10 or

TABLES I, II or VII. Notice that M has a cyclic normal subgroup Zm with factor group of odd prime order r such that r < logq m. Inspecting the candidates in Lemmas 4.3.3-

47 4.4. Subgroups of linear groups

4.3.10, TABLES I, II and VII, we obtain three possibilities: soc(K) = PSUr(q1) and

M = r : , or K = PSL (7) and M = : , or K = PSU (5).3 and Z(q1 +1)/(q1+1)(r,q1+1) Zr 2 Z7 Z3 3 M = : . Suppose that soc(K) = PSU (q ) and M = r : . If Z21 Z3 r 1 Z(q1 +1)/(q1+1)(r,q1+1) Zr

(q, q1) = 1, then it is easily shown that |K|q1 does not divide |T |, which is not possible.

Thus, (q, q1) 6= 1, from which it follows that q = q1, and so soc(K)=T , a contradiction. If

K = PSL2(7) and M = Z7:Z3, or K = PSU3(5).3 and M = Z21:Z3, then T = PSL3(4). By the Atlas [19], the case (T, M, K) = (PSL3(4), Z7:Z3, PSL2(7)) does occur, as shown 0 in TABLE III . The case K = PSU3(5).3 is impossible, as PSL3(4) contains no subgroup isomorphic to PSU3(5).3. Therefore if (n, q) 6= (3, 4), then M is maximal.

Case 2. Assume that T = PSL3(q). By Lemmas 4.3.3-4.3.10, M ∈ Ci for i = 1, 2, 3, or

6. If M ∈ C3, then M is maximal in T as shown in Case 1. Suppose that M ∈ C1. Then

1 [q3]: 2 =∼ M < K = P =∼ [q3]: GL(2, q) =∼ P Z(q−1)/(3,q−1) 1 (3, q − 1) 2

Let σ be the transpose-inverse. Then σ interchanges P1 with P2, so P1 and P2 are fused by σ in T.hσi, and a novelty maximal subgroup M.hσi of G = T.hσi arises. Furthermore, if q is a square, then there is a ﬁeld automorphism, say τ, of order 2.

This τ normalizes both P1 and P2, so στ interchanges P1 with P2, consequently M.hστi is also a novelty maximal subgroup of G = T.hστi. For q = 3, by the Atlas [19] the 2 maximal subgroups 3 :2S4 form two classes, as in TABLE A. If M ∈ C2 ∪ C6, then by [58, Theorem 2.4, 2.5], M is maximal in T , as listed in TABLE III. In particular, for 2 q = p ≡ 1 (mod 9), the C6-subgroups M = 3 :2A4 form three classes, as in TABLE A.

Case 3. Let T = PSL4(q). For q = 2 or 3, we read our result from the Atlas [19]. So assume that q ≥ 4. Then by Lemmas 4.3.3-4.3.10, the only subgroup that needs to be 3 considered is M = [(q − 1) /(4, q − 1)].S4, which is a C2-subgroup. 3 For q = 4, M = 3 .S4 is contained in a maximal C8-subgroup K = PSU4(2) [93]. All maximal C8-subgroups of PSL4(4) which are isomorphic to PSU4(2) are conjugate [62, Prop 4.8.5]. Thus no novelty occurs, as indicated in TABLE III0.

We thus assume further that q ≥ 5. Since M is irreducible, we have K/∈ C1. If ∼ 2 K ∈ Ci for i = 2, 3 or 5, then K is of type GL2(q) o S2, or K = (PSL2(q ) × Zq+1).2, or r K = PSL4(q0).a with q = q0, r prime and a | 4, respectively. Thus, |M| does not divide |K|, which is not possible. 4 Assume that K ∈ C6. Then K = 2 .A6.a, where a = 1 or 2. Since |M| divides 2 |K|, we have q = 5. Thus, M = 4 :S4 is indeed contained in K (We are grateful to C.Roney-Dougal who conﬁrmed this fact with the assistance of computers). Let Mˆ be the preimage of M in SL4(5). Then Mˆ consists of all matrices in SL4(5) which have only one nonzero element in each row and column, the ﬁrst three are freely chosen, and the

48 4.4. Subgroups of linear groups

last one is chosen such that the product of the four elements is 1. Now the non-trivial

outer automorphisms of T are induced by the diagonal matrices dλ := diag{λ, 1, 1, 1}, where λ∈ / {0, 1}. Let µλ denote the outer automorphism induced by dλ. Then simple calculation shows that these outer automorphisms all normalize, but do not centralize,

M. Suppose that µλ normalizes K. Then since µλ does not centralize M, we obtain 4 4 that hK, µλi = 2 .A6.hµλi, and A6.hµλi ≤ Aut(2 ) = GL4(2) = A8, which is not 2 possible. Thus µλ does not normalize K. It follows that M = 4 :S4.hµλi is a novelty

maximal subgroup of T.hµλi = PGL4(5). Let σ be the transpose-inverse. Then clearly 2 σ normalizes M, therefore M = 4 :S4.hµλσi is a novelty maximal subgroup of T.hµλσi, as shown in TABLE III and TABLE III0. So we next assume that q > 5. ± 1/2 If K ∈ C8, then K is of type Sp4(q), Ω4 (q), or SU4(q ), which is not possible as |M| 6 | |K|.

Thus, we assume that each overgroup of M is a C9-group. For a minimal overgroup K of M, the factor group K := K/Z(K) is almost simple and has a maximal soluble subgroup M := MZ(K)/Z(K). Thus, the pair (soc(K), M) appears in Lemmas 4.3.3- 4.3.10 and TABLE I, II or VII. Note that q > 5 and M has an abelian normal subgroup of rank 3 with the factor group S4. An inspection of the candidates in Lemmas 4.3.3- 4.3.10 and TABLES I, II and VII leads to the fact that (K, M) is one of the following 3 3 two pairs: (PSL4(q1), [(q1 − 1) /(4, q1 − 1)].S4), or (PSU4(q1), [(q1 + 1) /(4, q1 + 1)].S4). 3 Suppose that (K, M) = (PSL4(q1), [(q1 − 1) /(4, q1 − 1)].S4) with (q1, q) = 1, or 3 (K, M) = (PSU4(q1), [(q1 + 1) /(4, q1 + 1)].S4) with (q1, q) = 1. Then it is easily shown that |K|q1 does not divide |T |, which is a contradiction. Thus, (K, M) = 3 (PSL4(q1), [(q1 − 1) /(4, q1 − 1)].S4) with (q1, q) 6= 1. It then follows that q1 = q, and K = T , which is again a contradiction. So for q ≥ 7, M is a maximal subgroup of T , as listed in TABLE III.

Case 4. Finally, let T = PSLn(q), n ≥ 5. For (n, q) = (5, 2), the result follows from the Atlas [19]. Hence assume (n, q) 6= (5, 2). By Lemmas 4.3.3–4.3.10, we have one of the following:

8 2 (a) T = PSL5(3), and M = [3 ]:GL2(3) ,

12 3 3 (b) T = PSL6(2), and M = [2 ]:SL2(2) , or M = PSL2(2) .S3;

12 3 2 3 2 (c) T = PSL6(3), and M = [3 ]:SL2(3) , or M = 2 .PSL2(3) .2 .S3;

4 (d) T = PSL8(2), and M = PSL2(2) .S4;

3 4 3 (e) T = PSL8(3), and M = 2 .PSL2(3) .2 .S4.

49 4.4. Subgroups of linear groups

For the case (a), M is of type P2,3, thus M is contained in the maximal parabolic subgroups P2 and P3. If G contains a transpose-inverse automorphism σ, then σ ﬁxes 0 P2,3, and interchanges P2 and P3, giving rise to a novelty NG(M), as in TABLE III . 12 3 Similarly, for the cases T = PSL6(2) and M = [2 ]:SL2(2) , or T = PSL6(3) and 12 3 M = [3 ]:SL2(3) , if G contains a transpose-inverse σ, then NG(M) occurs as a novelty.

In particular, for T = PSL6(3), let δ be a diagonal automorphism. Then δ normalizes 0 P2,3, P2 and P3. Thus M.hδσi is a novelty of T.hδσi, as listed in TABLE III . n/2 Assume that T = PSLn(2) with n = 6 or 8, and M = PSL2(2) .Sn/2. First we consider the GL6(2)-geometry. Let V be the natural GL6(2)-module. Then the preimage

Mˆ of M in GL6(2) is the stabilizer of a 2-decomposition V = V1 ⊕ V2 ⊕ V3, where Vi are

2-dimensional subspaces of V . Since GL2(2) = Sp2(2), we conclude that M is contained in a C8-subgroup of type Sp6(2), which is isomorphic to K = PSp6(2). Thus M is not maximal in T . By the Atlas [19], M is contained in K1 = PSU4(2).2, which is a maximal subgroup of PSp6(2). Also by the Atlas [19], M has a unique conjugacy class in K1, so M has a unique conjugacy class in K. Since K has a unique conjugacy class in T , by

Lemma 4.3.13, no novelty occurs in Aut(T ) = PΓL6(2). Analogously it is easy to prove that for the case n = 8, M is contained in K = PSp8(2), and no novelty occurs, as listed in TABLE III0. 3 4 3 Suppose that T = PSL8(3) and M = 2 .PSL2(3) .2 .S4. Then by deﬁnition, in this case C5 and C7 are void. Since M is irreducible, we have K/∈ C1. Also because the 4 3 preimage Mˆ of M in GL8(3) is isomorphic to SL2(3) .[2 ].S4, and Mˆ is the stabilizer in

SL8(3) of the decomposition V = V1 ⊕ · · · ⊕ V4, where Vi is a 2-dimensional subspace of V , the group Mˆ consists of all matrices in SL8(3) that are composed of 2 × 2 block matrices, of which each row and column can contain only one nonezero block matrix in

GL2(3). It follows that K/∈ C2.

If K ∈ C3, then K is almost simple and soc(K) = PSL4(9), but then |M| does not divide |K|, which is a contradiction. If K ∈ C4, then M is contained in a subgroup 2 isomorphic to (PSL2(3)×PSL4(3)).[2 ], hence |M| - |K|, again a contradiction. Similarly, if K ∈ C6 ∪ C8, then calculation shows that |M| - |K|, which is impossible.

Thus K ∈ C9. Choose K to be a minimal overgroup of M. Then K := K/Z(K) is almost simple, and contains a soluble maximal subgroup M = MZ(K)/Z(K). Hence the pair (soc(K), M) appears in Lemmas 4.3.3–4.3.10 or TABLES I, II or VII, an inspection shows that no such pair exists, a contradiction. Thus M is a maximal subgroup of T , listed in TABLE III. 2 3 2 Analogously, for T = PSL6(3) and M = 2 .PSL2(3) .2 .S3, M is a maximal subgroup of T .

50 4.5. Subgroups of symplectic groups

4.5 Subgroups of symplectic groups

Let soc(G) = T = PSpn(q), where n ≥ 4. For G = PSp4(q), the maximal subgroups of G have been determined (see [91] or [58, Theorem 2.8] for q odd, and [35] for q f even). For G = PSp4(2 ).hσi, where σ is a graph automorphism, most of the maximal subgroups of G do not belong to the standard Aschbacher classes, but have diﬀerent structures. Therefore Aschbacher introduced some alternative classes to describe them, see [1, Section 14]. Also for the general case, the maximality of certain Ci-subgroups is already known [58].

Proposition 4.5.1. Let T = PSpn(q) with n ≥ 4, and let M be a soluble maximal Ci- subgroup of T . Then either M is a maximal subgroup of T , and (T,M) = (G0,H0) lies in TABLE IV, or M < K < T is such that (T,M,K) lies in TABLE IV0. In particular, there is only one conjugacy class of such subgroups H0, and no novelty occurs.

TABLE IV0

TMK

PSp4(3) 2.PGU2(3).2 2.(A4 × A4).2 PGL2(3).2 2.(A4 × A4).2 3 − PSp6(2) S3.S3 O6 (2) 2 PSp6(3) A4 × S4 [2 ].(A4 o S3) 4 + PSp8(2) S3.S4 O8 (2):2 + 3 2 PSp8(3) (PSp2(3) × PO4 (3)).2 PSp2(3) .2 .S3

Proof. As mentioned before the proposition, maximal subgroups of G for n = 4 are f known. For G = PSp4(2 ).hσi, σ a graph automorphism, the maximal subgroups of G are listed in [50, Theorem 12.1], from which we found the following soluble maximal 4 2 subgroups of G:[q ]:Zq−1.hσi, (D2(q±1) o 2).hσi and (Zq2+1.4).hσi. All of them are

novelties, as indicated in TABLE IV (when q = 4, the novelty subgroup (D2(q−1) o 2).hσi

does not occur). By Lemmas 4.3.3–4.3.10, there leaves only two cases: T = PSp4(2)

and T = PSp4(3). For the latter, the result follows from the Atlas [19], while the former 0 ∼ is omitted since PSp4(2) = PSL2(9). Thus, we assume that n ≥ 6. By Lemmas 4.3.3–4.3.10, for symplectic groups with

n ≥ 6, there are only a few candidates: T = PSpn(q), where n = 6 or 8, and q = 2 or 3. For the cases (n, q) = (6, 2), (6, 3) and (8, 2), the result follows from the Atlas [19], as listed in TABLE IV and TABLE IV0. Thus we only need to consider the case (n, q) = (8, 3).

Suppose that T = PSp8(3). Then by Lemmas 4.3.3–4.3.10, M is a C2-subgroup 3 + 2 .(PSp2(3) o S4) or a C4-subgroup (PSp2(3) × PO4 (3)).2. 3 First we show that the C2-subgroup M = 2 .(PSp2(3) o S4) is maximal in T . Suppose

that a subgroup K of T is such that M < K < T . Then in this case the classes C5 and C7

51 4.5. Subgroups of symplectic groups

are empty. It follows from Lagrange’s Theorem that K/∈ C1 ∪ C3 ∪ C4 ∪ C6 ∪ C8. Suppose that K ∈ C2. Since |M| does not divide |GL4(3).2|, K is a subgroup of type Sp4(3) o Z2, which is the stabilizer of a non-degenerate decomposition V = V1⊥V2 in Sp8(3), where dim(Vi)=4. Consider the decompositions V1 = V11⊥V12 and V2 = V21⊥V22, where ˆ dim(Vij)=2. There is an element g in the preimage M of M in Sp8(3) such that g

ﬁxes V12 and V21, and interchanges V11 and V22. Thus g does not ﬁx the decomposition

V = V1⊥V2, and so g∈ / Kˆ and M 6< K.

Therefore, each overgroup of M is almost quasisimple of C9-type. For a minimal overgroup K of M, the factor group K := K/Z(K) is almost simple and has a maximal soluble subgroup M := M/M ∩Z(M). So the pair (soc(K), M) appears in Lemmas 4.3.3- 4.3.10, or TABLES I, II or VII. But an inspection of the candidates shows that no such 3 pair exists, a contradiction. Thus M = 2 .(PSp2(3) o S4) is maximal in PSp8(3). + Next, suppose that M = (PSp2(3) × PO4 (3)).2, a C4-subgroup. We show that M is not maximal in T . Let V be the natural Sp8(3)-module of dimension 8. Then the ˆ preimage M of M in Sp8(3) is the stabilizer of a tensor decomposition V = V1 ⊗ V2, where (V1, f1) is a symplectic geometry of dimension 2, and (V2, f2) is an orthogonal geometry of dimension 4, with fi being a non-degenerate bilinear form on Vi. It turns out + that the orthogonal geometry (V2, f2) (the O4 (q)-geometry) has a tensor decomposition

V2 = W1 ⊗ W2, where for some symplectic forms hi,(Wi, hi) is a symplectic geometry of dimension 2. Thus f1 ⊗ h1 ⊗ h2 is a symplectic form on V , and M stabilizes the tensor decomposition V = V1 ⊗ (W1 ⊗ W2) = V1 ⊗ W1 ⊗ W2. Since the stabilizer of ˆ 3 this decomposition is a subgroup isomorphic to K := Sp2(3) .S3, we have M < K, where K is the image of Kˆ in T . Moreover, subgroups of T which are isomorphic to M or K form a unique conjugacy class in T . On the other hand, the only outer automorphisms of T are those that are induced by similarities, that is, elements g ∈ g g GL8(3) such that f(u , v ) = λgf(u, v), where f is the corresponding form and u, v ∈ V .

Since the ﬁeld is of order 3, the only scalar λg that corresponds to a non-trivial outer automorphism is λg = −1. Now the diagonal matrix D1 = diag{1, −1} induces a non- trivial outer automorphism of PSp2(3), because it corresponds to the scalar −1. For the same reason, D2 = diag{−1, 1, −1, 1} induces a non-trivial outer automorphism of + PΩ4 (3). Thus the Kronecker product D1 ⊗D2 induces a non-trivial outer automorphism of M. Since it corresponds to the scalar −1, it is a non-trivial outer automorphism of + PSp8(3). Note that Sp2(3) × Sp2(3) ≤ O4 (3), so D2 induces an outer automorphism of

PSp2(3) × PSp2(3). It follows that D1 ⊗ D2 normalizes K. Thus by Lemma 4.3.13, no 0 novelty occurs, as indicated in TABLE IV .

52 4.6. Subgroups of unitary groups

4.6 Subgroups of unitary groups

In the literature, maximal subgroups of unitary groups have been determined in some special cases. The maximal subgroups of PSU3(q) were determined by Mitchell [90] and

Hartley [48] (refer to [58, Theorem 2.6, 2.7]). For G = ΓU4(q) or ΓU5(q), subgroups

of G were described in [57]. For certain i, the maximality of the maximal Ci-subgroups

of PSUn(q) is known [58]. In the general case, soluble maximal subgroups of unitary groups are determined in the following proposition.

Proposition 4.6.1. Let T = PSUn(q) with n ≥ 3, and let M be a soluble maximal

Ci-subgroup of T . Then either M is a maximal subgroup of T , and (T,M) = (G0,H0) lies in TABLE V, or M < K < T is such that the triple (T,M,K) lies in TABLE V0. 0 Moreover, the group G0 in TABLE V is such that T < G0 ≤ Aut(T ), G0 contains a novelty H0, and H0 ∩ T = M. In particular, with the exceptions listed in TABLE A, all such subgroups H0 are conjugate.

TABLE V0

TMKG0 H0 PSU3(3) 7:3 PSL2(7) − − S4 PSL2(7) − − PSU3(5) 7:3 A7 PGU3(5) 7:3 × 3 2 [12] : S3 A7 PGU3(5) 6 :S3 2 2 3 :Q8 M10 PGU3(5) 3 : 2A4 2 3 PSU4(2) 3.S3.2 3 .S4 − − 4 4 2 PSU4(3) [2 ].S4 2 :A6 T.2 (4 × 2).S4 4 4 3 [2 ].S4 2 :A6 PGU4(3) 4 .S4 PSU5(2) 11:5 PSL2(11) − − 2 5 2 PSU6(2) S3 × (3 :Q8) [3 ].Q8:S3 − − 3 3.S3.S3 PSU4(3):2 − − 3 4 7 PSU8(2) [3 ].S3.S4 3 .S8 − −

Proof. By Lemmas 4.3.3–4.3.10, there are several possibilities for T and M: (a) T is one

of a few small groups; (b) T = PSUr(q) with r prime, and M ∈ C3; (c) T = PSU3(q)

with M ∈ C1 ∪ C2 ∪ C5 ∪ C6; (d) T = PSU4(q), and M ∈ C2. For the cases where n = 3 and 3 ≤ q ≤ 11, or (n, q) = (4, 2), (4, 3), (5, 2) or (6, 2),

the statement follows from the Atlas [19]. In particular, for T = PSU3(8), subgroups 2 isomorphic to 3 :2A4 form three classes, as in TABLE A. Thus, assume that (n, q) is not one of these pairs. 3 For part (d), T = PSU4(q), and M = [(q + 1) /(4, q + 1)].S4 ∈ C2 with q ≥ 4, and by [27], M is maximal in T , as listed in TABLE V.

Assume that T = PSU3(q) with q > 11, and M ∈ C1 ∪ C2 ∪ C5 ∪ C6, as in part (c). 3 By Lemmas 4.3.3 and 4.3.4, M is isomorphic to [q ]:Z(q2−1)/(3,q+1) ∈ C1, or (Zq+1 ×

53 4.6. Subgroups of unitary groups

r Z(q+1)/(3,q+1)):S3 ∈ C2; or by Lemmas 4.3.7 and 4.3.7, either T = PSU3(2 ) with r ≥ 3 2 prime, and M = PSU3(2).(3, r) = 3 :Q8.(3, r) ∈ C5, or T = PSU3(q) with q = p ≡ 2 2(mod 3), and M = 3 :Q8.a ∈ C6, with a = 1 or 3. For each of these cases, by the results of Mitchell and Hartley (see [58, Theorem 2.6, 2.7]), M is maximal in T , as in

TABLE V. Furthermore, for T = PSU3(q) and q = p ≡ 8 (mod 9), the C6-subgroups 2 2 M = 3 :Q8.3 = 3 :2A4 form three classes, as in TABLE A. For parts (a) and (b), suppose that M is not maximal in T , and let M < K < T .

∼ Case 1. Let T = PSUr(q) with r an odd prime, and M = Zm:Zr ∈ C3, where m = qr+1 (q+1)(r,q+1) . Since (r, q) 6= (3, 2), by Theorem 4.3.14, q2r − 1 has a primitive prime divisor t say, r which divides q + 1. Since M is irreducible, so is K, hence K/∈ C1. As M ∈ C3, by

Lemma 4.3.11, K/∈ C3. Noticing that the dimension r is a prime, there is neither a C4 r nor a C7-subgroup in T . If K ∈ C2, then |K| divides |GU1(q) o Sr| = (q + 1) r!, which is not divisible by t, and this is not possible. If K ∈ C5 ∪ C6 ∪ C8, then K = GUr(q0) with e 2 q = q0 and e > 1, or r .Sp2(r), or Ωr(q), respectively; however, t does not divide |K|, a contradiction.

Thus, by Lemma 4.3.11, each overgroup of M is a C9-type almost quasisimple group. Let K be a minimal overgroup of M in T . Then K := K/Z(K) is almost simple, and contains a soluble maximal subgroup M := MZ(K)/Z(K). Hence the pair (soc(K), M) appears in Lemmas 4.3.3–4.3.10, or TABLES I, II or VII. Notice that M has a normal cyclic subgroup Zm with factor group Zr, such that the odd prime r < logq m. An inspection of Lemmas 4.3.3–4.3.10 and TABLES I, II and VII shows that all possible pairs (K, M) are as follows: (soc(K), M) = (PSL (q ), r : ), r 1 Z(q1 −1)/(q1−1)(r,q1−1) Zr or (PSU (q ), r : ), or (PSL (4), : ), or (PSL (11), : ), or r 1 Z(q1 +1)/(q1+1)(r,q1+1) Zr 3 Z7 Z3 2 Z11 Z5 (PSL2(7), Z7:Z3), or (A7, Z7:Z3). For the ﬁrst two cases, it is easily shown that either |K|q1 does not divide |T |, or q1 = q and soc(K) = PSLr(q), which is not possible. By Proposition 5.1, the subgroup Z7:Z3 is not maximal in PSL3(4), thus the case (soc(K), M) = (PSL3(4), Z7:Z3) is eliminated. The case (soc(K), M) = (PSL2(11), Z11:Z5) corresponds to T = PSU5(2), the case (soc(K), M) = (PSL2(7), Z7:Z3) corresponds to T = PSU3(3), while the case (soc(K), M) = (A7, Z7:Z3) corresponds to T = PSU3(5). By our assumption, all are impossible. Thus M is a maximal subgroup of T as listed in TABLE V.

Case 2. Here we deal with the remaining cases which were stated in part (a). Except for 5 groups that appear in the Atlas [19], these cases involve the following small groups:

(q+1)t−1(q+1,2) t t−1 (1) T = PSU2t(q), and M = [ (q+1,2t) ].PSU2(q) .[(q+1, 2) ].St, where (2t, q) = (6, 3), (8, 2) or (8, 3).

54 4.6. Subgroups of unitary groups

t−1 t t−1 (2) T = PSU3t(2), and M = [3 ].PSU3(2) .[3 ].St, where t = 3 or 4.

3 4 For the case (1), suppose that T = PSU8(2) = SU8(2) and M = [3 ].S3.S4. First we ∼ 4 consider the GU8(2)-geometry. Then the preimage Mˆ = GU2(2) .S4 of M in GU8(2) is the stabilizer of a decomposition of V into 2-dimensional subspaces

V = V1 ⊕ V2 ⊕ V3 ⊕ V4.

Next we consider the GU2(2)-geometry. Let {xi, yi} be an orthonormal basis of Vi, where

1 ≤ i ≤ 4. Then hxii and hyii are the only non-singular 1-spaces of Vi. Thus GU2(2) ﬁxes a unique non-degenerate 1-subspace decomposition. It follows that Mˆ is contained in a 8 7 larger C2-subgroup isomorphic to Kˆ := GU1(2) .S8. Thus M is contained in K = 3 .S8, the image of Kˆ in T . Further, both M and K have a unique conjugacy class in T . Let τ be a non-trivial outer automorphism of T . Then τ is a ﬁeld automorphism of order

2, and clearly τ normalizes each C2-subgroup, so τ normalizes both M and K. Thus no novelty occurs. The triple (T,M,K) is listed in TABLE V’. 2 3 2 Assume that T = PSU6(3) and M = [4 ].A4.[2 ].S3, or T = PSU8(3) and M = 5 4 3 [2 ].A4.[2 ].S4. We aim to show that M is maximal in T , as listed in TABLE V. We consider the overgroup K of M. Since M is absolutely irreducible, we obtain that K/∈ C1 ∪ C3. If K ∈ C4 ∪ C5 ∪ C6, then it is easily shown that |M| does not divide

|K|, a contradiction. If K ∈ C2, then since q = 3 is a prime, the subgroup M is not contained in a C2-subgroup of type GU1(q) o Sn. Thus K is of type GLn/2(9).2, where n = 6 or 8. However in both cases |M| does not divide |K|, again a contradiction.

Since the classes C7 and C8 do not appear, each overgroup of M is almost quasisimple. Choose K to be a minimal overgroup of M. Then K := K/Z(K) is almost simple, and 2 3 2 contains a soluble maximal subgroup M := MZ(K)/Z(K). Since M = [4 ].A4.[2 ].S3 or 5 4 3 [2 ].A4.[2 ].S4, an inspection of Lemmas 4.3.3–4.3.10 and TABLES I, II and VII shows that no such candidate exists, a contradiction. Thus M is maximal in T . t−1 t t−1 Finally, we consider the cases T = PSU3t(2), and M = [3 ].PSU3(2) .[3 ].St,

where t = 3 or 4. Suppose that M has an overgroup K. Then by deﬁnition K/∈ C8.

Since M is absolutely irreducible, we obtain that K/∈ C1 ∪C3. Calculation on the orders

shows that K/∈ C2 ∪ C4 ∪ C5 ∪ C6 ∪ C7. This leaves the only case of K ∈ C9. We choose K to be a minimal overgroup of M. Let K = K/Z(K) and M = MZ(K)/Z(K). Then M is a soluble maximal subgroup of K. By inspecting Lemmas 4.3.3-4.3.10 and TABLES I, II and VII, we found no such pair K and M, a contradiction. Thus M is a maximal

subgroup of T as listed in TABLE V.

55 4.7. Subgroups of orthogonal groups

4.7 Subgroups of orthogonal groups

We here deal with subgroups of orthogonal groups. We ﬁrst prove a simple lemma.

Lemma 4.7.1. Let V be a 3-dimensional natural O3(3)-module for G = O3(3). Then G stabilizes a non-degenerate isometric 1-decomposition V = V1⊥V2⊥V3, where dim Vi = 1.

Proof. Let Q be a non-degenerate quadratic form deﬁned on V , and (u, v) be the associated bilinear form. Then there exists a basis {u1, u2, u3} of V such that V = hu1i⊥hu2i⊥hu3i is a non-degenerate isometric 1-decomposition, with (ui, ui) = 1 or

−1. The stabilizer of this decomposition in G is O1(3) o S3. Since O1(3) = GL1(3), 3 4 we have O1(3) o S3 = 2 :S3, the order of which is 2 .3, and equals to |O3(3)|. Thus

O3(3) = O1(3) o S3 stabilizes this 1-decomposition.

+ For T = PΩ8 (q), up to conjugacy, all the maximal subgroups of G are determined by Kleidman in [59], from which we read oﬀ our result. By Lemmas 4.3.3–4.3.10, if

T = PΩ7(q), then q = 3 and M is a C1-subgroup, which lies in the Atlas [19]. Generally, the maximality of Ci-subgroups of orthogonal groups are determined as follows.

Proposition 4.7.2. Let T be an orthogonal simple group and let M be a soluble maximal

Ci-subgroup of T . Then either M is a maximal subgroup of T , and (T,M) = (G0,H0) lies in TABLE VI, or M < K < T is such that the triple (T,M,K) lies in TABLE VI0. 0 Moreover, the group G0 in TABLE VI is such that T < G0 ≤ Aut(T ), G0 contains a novelty H0, and H0 ∩ T = M. Further, there is only one conjugacy class of such subgroups H0. TABLE VI0

TMKG0 H0 + 3 − 4 3 PΩ8 (2) [2 ].S4 Ω2 (2) .2 .S4 − − + 2 − 4 3 Ω4 (2) .2.2 Ω2 (2) .2 .S4 − − + 2 + 4 3 9 PΩ8 (3) 2 .Ω2 (3) .2 .S4 [2 ].S4 − − 9 6 2 2 [2 ].S4 [2 ].A8 T.2 M.2 9 6 [2 ].S4 [2 ].A8 T.4 M.4 + 4 3 + PΩ8 (4) 3 .2 .S4 PΩ8 (2) − − + 9 6 2 2 PΩ8 (5) [2 ].S4 [2 ].A8 T.2 M.2 9 6 [2 ].S4 [2 ].A8 T.4 M.4 2 3 2 8 Ω9(3) (2 × Ω3(3) .2 ).S3 2 .A9 − − + + 3 2 − 6 5 PΩ12(2) Ω4 (2) .2 .S3 Ω2 (2) .2 .S6 − − + + 2 + 3 4 PΩ12(3) PO4 (3) ◦ SO3(3) 2 .PΩ4 (3) .2 .S3 − − 2 4 3 10 2 × Ω3(3) .2 .S4 2 .A12 − − + + 4 3 − 8 7 PΩ16(2) Ω4 (2) .2 .S4 Ω2 (2) .2 .S8 − −

56 4.7. Subgroups of orthogonal groups

Proof. As mentioned before the proposition, maximal subgroups for T = PΩ7(q) and + T = PΩ8 (q) are known. Thus, assume that n ≥ 9. By Lemmas 4.3.3–4.3.10, there are a + few possibilities for T to have soluble maximal Ci-subgroups: T = Ω9(3), or T = PΩ8 (q), + or T = PΩ4t(p), where t = 2, 3 or 4, and p = 2 or 3. 2 3 2 Assume that T = Ω9(3) and M is a C2-subgroup (2 × Ω3(3) .2 ).S3 of type O3(3) o

S3. Let V be the natural O9(3)-module. Then the preimage Mˆ of M in O9(3) is the stabilizer of a 3-decomposition V = V1⊥V2⊥V3, where Vi are non-degenerate isometric subspaces of dimension 3. We consider the O3(3)-geometry. By Lemma 4.7.1, the group O3(3) stabilizes a non-degenerate isometric 1-decomposition Vi = Vi1⊥Vi2⊥Vi3, where 1 ≤ i ≤ 3. Thus Mˆ stabilizes the non-degenerate isometric 1-decomposition

V = V11⊥V12⊥ ... ⊥V33, which means that Mˆ is contained in a C2-subgroup of type

O1(3) o S9. Thus M is not maximal. Also because Aut(T ) = PO9(3) and |Out(T )| = 2, we conclude that no novelty occurs, as shown in TABLE VI0. + + 3 2 Assume that T = PΩ12(2), and M = Ω4 (2) .2 .S3 is a C2-subgroup. Let V be ˆ ∼ + 3 the natural O12(2)-module. Then the preimage M = O4 (2) .S3 is the stabilizer of + a 4-decomposition V = V1 ⊕ V2 ⊕ V3 in O12(2), where Vi is a 4-dimensional natural + + + O4 (2)-module. We consider the O4 (2)-geometry. For each i,O4 (2) ﬁxes a (−2)- space decomposition Vi = Vi1⊥Vi2, where Vij is a non-degenerate isometric space of dimension 2, and the stabilizer of this (−2)-space decomposition of Vi has the structure − − ˆ ˆ O2 (2) × O2 (2). It follows that M stabilizes a (−2)-space decomposition of V . Thus M − is contained in one of the C2-subgroups of type O2 (2) o S6. Therefore, M is not maximal + in T . Since Aut(T ) = O12(2) and |Out(T )| = 2, we conclude that no novelty arises in + 0 O12(2), as in TABLE VI . + 2 4 3 2 + 3 4 + Assume T = PΩ12(3), and M = 2 ×Ω3(3) .2 .S4, or 2 .(PΩ4 (3)) .2 .S3, or PΩ4 (3)×

SO3(3). + Suppose that M is a C4-subgroup isomorphic to PΩ4 (3) × SO3(3). By Lemma 4.7.1,

O3(3) fixes a non-degenerate isometric 1-decomposition, so M is contained in a subgroup + K of type O4 (3)oS3. Thus M is not maximal in T . First note that both M and K form a unique conjugacy class in T . Now Out(T ) = D8, the outer automorphisms of T which are + 2 2 induced from Ω12(3) is a normal 4-subgroup Z2. Thus if G = T.O and O ≤ Z2, then no novelty occurs. Now we consider the outer automorphisms induced by similarities, that g g is, those induced by elements g ∈ GL12(3) such that f(u , v ) = λgf(u, v). It follows since the field is of order 3 that λg = −1. Since D = diag{−1, 1, −1, 1} induces a non- + trivial outer automorphism of PΩ4 (3), D ⊗ I3 is non-trivial and fixes M. Furthermore, 0 D ⊗ I3 fixes K. Thus by Lemma 4.3.13, no novelty occurs, as in TABLE VI . 2 4 3 Similarly, if M is a C2-subgroup 2 × Ω3(3) .2 .S4, then M is contained in a C2- 10 subgroup K of type O1(3) o S12, where by [62, Prop 4.2.15], K is isomorphic to 2 .A12,

57 4.8. Tables and no novelty occurs. ∼ 2 + 3 4 In the case M = 2 .(PΩ4 (3)) .2 .S3, we show that M is maximal in T . Suppose that there exists a subgroup K of T such that M < K < T . Since M is absolutely irreducible on V , K/∈ C1 ∪ C3. Comparing the orders we know that K/∈ C2 ∪ C4. Since q = 3 and the dimension is 12, the classes C5, C6, C7 and C8 are void. Thus K ∈ C9. As before, we choose K to be a minimal overgroup of M. Then K := K/Z(K) is almost simple, and contains a soluble maximal subgroup M = MZ(K)/Z(K). Hence the pair (soc(K), M) appears in Lemmas 4.3.3–4.3.10, or TABLES I, II or VII, an inspection shows that no such pair can be found, a contradiction. Thus, M is maximal, and (T,M) = (G0,H0) lies in TABLE VI. + + 4 3 Assume that T = PΩ16(2). Then the only possibility is M = Ω4 (2) .2 .S4, a C2- − subgroup. By [62, TABLE 3.5H], M is contained in a C2-subgroup of type O2 (2) o S8 (with only one conjugacy class in T ), so M is not maximal, as shown in TABLE VI’. + 3 + 4 6 Finally, let T = PΩ16(3). Then M = 2 .PΩ4 (3) .2 .S4 is a C2-subgroup. By [62, TABLE 3.5E], M is maximal in T as listed in TABLE VI.

4.8 Tables

In the last section we present a complete list of the soluble maximal subgroups of almost simple groups in seven tables: TABLE I– TABLE VII. First we make some conventions.

Remarks on the tables: (1). As indicated in Theorem 4.1.1, for an almost simple group G and a soluble maximal subgroup H of G, a normal subgroup G0 of G is chosen to be minimal subject to the condition that H0 := H ∩ G0 is maximal in G0, and H = H0.(G/G0). ∼ (2). For ﬁnite simple groups we have the following generic isomorphisms: A5 = ∼ ∼ ∼ ∼ 0 ∼ PSL2(4) = PSL2(5), PSL2(7) = PSL3(2), A6 = PSL2(9) = PSp4(2) ,A8 = PSL4(2), ∼ ∼ 0 ∼ 0 PSp4(3) = PSU4(2), PSL2(8) = Ree(3) , and PSU3(3) = G2(2) . For convenience we listed all the soluble maximal subgroups of these isomorphic groups in the tables, except 0 for the groups with socle PSp4(2) . In this case the list is too long and double-listing is omitted, and the readers are suggested to get the information of the maximal subgroups from that of PSL2(9) given in Table III. (3). As indicated in Theorem 4.1.1, for an almost simple group G, most of the soluble maximal subgroups of G have only one conjugacy class in G, with only a few exceptions. For those that have more than one conjugacy class, the numbers of the conjugacy classes are 2 or 3, as shown in TABLE A of Theorem 4.1.1. A maximal subgroup is listed k times in a table of TABLE I– TABLE VII if it has k conjugacy classes.

58 4.8. Tables

f f f (4). Assume that T : = soc(G) = PSp4(2 ), G2(3 ) or F4(2 ). Then T admits ∼ a graph automorphism, and Out(T ) = Z2f . In each case when G contains a graph automorphism, then most of the (soluble) maximal subgroups of G are novelties. Assume ∼ that Out(T ) = hxi = Z2f . Then x squares to a generating ﬁeld automorphism. If f ∼ is odd, then Out(T ) = Zf × Z2, and so any involution in Aut(T )\T induces a graph f automorphism. Let µ = x . Then µ is a graph automorphism of order 2. Since G0 ∼ k is minimal, in this case we choose G0 = T.hµi = T.2. If f is even, say f = 2 m, let m ∼ µ = x . Then µ is a graph automorphism, and we choose G0 = T.hµi = T.Z2k+1 . The graph automorphism µ appears in TABLE IV and TABLE VII.

59 4.8. Tables

TABLE I. Soluble maximal subgroups of An and Sn.

G0 H0 A5 A4, S3 2 A6 3 :4, S4, S4 A7 (A4 × 3):2 4 A8 2 :(S3 × S3) 4 S8 2 :S4 3 2 A9 3 :S4, 3 :2A4 6 3 4 3 A12 2 :3 :S4, 3 :2 :S4 A16 (S4 o S4) ∩ G0 Ap Zp:Z p−1 , p prime, p 6= 7, 11, 17, 23 2 Sp Zp:Zp−1, p prime, p = 7, 11, 17, 23

60 4.8. Tables

TABLE II. Soluble maximal subgroups of sporadic groups.

G0 H0 2 M11 3 :Q8.2, 2S4 2 1+4 2 2 M12 A4 × S3, 4 :D12, 2+ .S3, 3 :2S4, 3 :2S4 1+2 M12.2 3+ :D8 M23 23:11 3 J1 2 :7:3, 19:6, 11:10, D6 × D10, 7:6 2+4 2 J2 2 :(3 × S3), 5 :D12 2+4 2 2 J3 2 :(3 × S3), 3 .(3 × 3 ):8 J3.2 19:18 1+2 J4 11+ :(5 × 2S4), 29:28, 43:14, 37:12 1+2 5 Ru 5+ :[2 ] 2 7 2 Co3 2 .[2 .3 ].S3 1+2 Co2 5+ :4S4 2 2 3 4 2 6 3+4 2 Co1 7 :(3 × 2A4), 3 .[2 .3 ].2A4, 3 .[2.3 ].2A4, 3 :2.S4 Ly 67:22, 37:18 9 2 7 1+2 2 Th [3 ].2S4, 3 .[3 ].2S4, 5+ :4S4, 7 :(3 × 2S4), 31:15 1+6 3+4 2 Fi22 3+ :2 :3 .2 1+8 1+6 1+2 Fi23 3+ .2− , 3+ .2S4 0 Fi24 29:14 1+2 Fi24 7+ :(6 × S3).2 1+2 5 HS.2 5+ :[2 ] 1+2 McL 5+ :3:8 2+4 2 McL.2 2 :S3 1+2 2 He 7+ :(S3 × 3), 5 :4A4 4+4 2 He.2 2 :3 :D8 2+4 2 Suz 3 :2.(A4 × 2 ).2 0 4 1+4 O N 3 :2− .D10 0 1+2 O N.2 31:30, 7 :(3 × D16) 1+4 1+4 4 2 HN 5+ :2− .5.4, 3 :2.A4.4 2 3 6 2 2 2 B 3 .3 .3 .(S4 × 2S4), (2 × 7 :(3 × 2A4)).2, (17 : 8 × 2 ).2, 13:12 × S4, 19:18 × 2, 23:11 × 2, 31:15, 47:23 1+2 M 13+ :(3 × 4S4), 23:11 × S4, (29:14 × 3).2, 31:15 × S3, 41:40, 47:23 × 2, 59:29, 71:35

61 4.8. Tables

TABLE III. Soluble maximal subgroups of linear groups

where q = pf , τ is the transpose-inverse, σ is a ﬁeld automorphism,

and δ is the diagonal automorphism

G0 H0 conditions PSL2(q) D2(q+1)/(2,q−1) q 6= 7, 9 D2(q−1)/(2,q−1) q 6= 5, 7, 9, 11 f Zp :Z(q−1)/(2,q−1) A4 q = p ≡ ±13, ±37(mod 40), or q = 5, or q = 3f , f ≥ 3 prime S4, S4 q = p ≡ ±1 (mod 8) PGL2(p) S4 p ≡ ±11, ±19 (mod 40) D12, D16 p = 7 D20 p = 11 PGL2(9) D16, D20 PSL2(9)·2 5:4, 8:2 G0 = M10 PSL3(2) 7:3, S4, S4 PSL3(2).hτi D12, D16 (q−1)2 PSL3(q) [ (3,q−1) ].S3 q 6= 2, 4 2 3 :Q8 q = p ≡ 4, 7 (mod 9) or q = 4 2 2 2 3 :Q8:3, 3 :Q8:3, 3 :Q8:3 q = p ≡ 1 (mod 9) 2 2 3 :Q8:S3, 3 :Q8:S3 q = 3 PSL3(3).hτi 2S4:2 3 (q−1)2 PSL3(q).hτi [q ]:[ (3,q−1) ].2 q > 2 3 (q−1)2 2f PSL3(q).hστi [q ]:[ (3,q−1) ].2 q = p , o(σ) = 2 3 (q−1)2 PGL3(q).hτi [q ]:[ (3,q−1) ].S3 3|(q − 1) PGL3(4) 7:3 × 3 (q−1)3 PSL4(q) [ (4,q−1) ].S4 q ≥ 7 4 PSL4(2) 2 :(S3 × S3) 4 PSL4(2).hτi 2 :S4 4 2 2 PSL4(3) 3 :2.A4.2, S4 2 PGL4(3) 2.S4.2 1+4 PSL4(3).hτi 3+ :(2S4 × 2) 2 1+4 PSL4(3).hδτi 2.S4.2, 3+ :(2S4 × 2) 3 PGL4(5) 4 .S4 3 PSL4(5).hδτi 4 .S4 8 2 PSL5(2).hτi [2 ]:S3.2 8 2 PSL5(3).hτi [3 ]:(2S4) .2 12 3 PSL6(2).hτi [2 ]:S3.2 2 3 2 PSL6(3) [2 ].A4.[2 ].S3 12 3 PSL6(3).hτi [3 ]:(2A4) .2.2 12 3 PSL6(3).hδτi [3 ]:(2A4) .2.2 3 4 3 PSL8(3) [2 ].A4.[2 ].S4 PSLr(q) Z(qr−1)/(q−1)(r,q−1):Zr r ≥ 3 prime, (r, q) 6= (3, 4)

62 4.8. Tables

TABLE IV. Soluble maximal subgroups of symplectic groups where µ is a graph automorphism deﬁned in Remark (4) on Page 24.

G0 H0 condition 4 2 PSp4(q).hµi [q ]:Zq−1.hµi, (D2(q±1) o S2).hµi, Zq2+1.4.hµi q > 4 even 3 2 3 PSp4(3) [3 ]:2A4, 2.A4.2, 3 :S4 2 2+4 2 5 PSp4(4).4 (2 × 2 :3):12, 17:16, 5 :[2 ] 6 2 PSp6(2) 2.[2 ]:S3 3+4 2+6 3 PSp6(3) 3 .2.(S4 × A4), 2 :3 :S3 3 PSp8(3) [2 ].(A4 o S4)

TABLE V. Soluble maximal subgroups of unitary groups where q = pf with p prime.

G0 H0 conditions 3 PSU3(q) [q ]:Z(q2−1)/(3,q+1) 2 [(q + 1) /(3, q + 1)].S3 q 6= 5 2 3 :Q8 q = p ≡ 2, 5 (mod 9), q 6= 5 2 2 2 3 :Q8.3, 3 :Q8:3, 3 :Q8:3 q = p ≡ 8 (mod 9) 2 2 2 PSU3(8) 3 :Q8.3, 3 :Q8:3, 3 :Q8:3 f 2 PSU3(2 ) 3 :Q8 f > 3 prime PSU3(3) 4.S4 2 2 PGU3(5) 7:3 × 3, 3 :2A4, 6 :S3 3 PSU4(q) [(q + 1) /(4, q + 1)].S4 q > 3 3 2 3 PSU4(2) [3 ]:2A4, 2.A4.2, 3 :S4 1+4 2 PSU4(3) 3+ :2S4, 2.A4.4 3 PGU4(3) 4 S4 2 PSU4(3).23 (4 × 2)S4 see [2] for the description of 23. 2 2 2 PSU4(3).(2 )133 (4 × 2)(2 × S4) see [2] for the description of (2 )133. 3 PSU4(3).D8 4 (2 × S4) 1+6 1+2 3 PSU5(2) 2− :3+ :2A4, S3 × [3 ]:2A4 5 2 PSU6(2) [3 ].Q8:S3 2 3 2 PSU6(3) [4 ].A4.[2 ].S3 5 4 3 PSU8(3) [2 ].A4.[2 ].S4 8 3 2 PSU9(2) [3 ].Q8.[3 ].S3 11 4 3 PSU12(2) [3 ].Q8.[3 ].S4 PSUr(q) Z(qr+1)/(q+1)(r,q+1):Zr r prime, (r, q) 6= (3, 3), (3, 5), (5, 2)

63 4.8. Tables

TABLE VI. Soluble maximal subgroups of orthogonal groups + where τ is a triality of PΩ8 (q), σ a ﬁeld automorphism

G0 H0 conditions 7 2 PΩ7(3) [3 ].(2A4 × A4).2, S4 × 2.A4.2 + 4 3 f PΩ8 (q) Zq−1.2 .S4 q = 2 ≥ 8 4 3 Zq+1.2 .S4 q even 4 3 3 (Z(q−1)/2.2 .2 .S4)/Z2 q ≥ 7 odd 4 3 3 (Z(q+1)/2.2 .2 .S4)/Z2 q ≥ 5 odd + 2 2 PΩ8 (q).hτi D2(q2+1)/(2,q−1).2 .3 q 6= 3 + 2 2 PΩ8 (q).hτσi D2(q2+1)/(2,q−1).2 .3 o(σ) = 3 + 2 2 2 PΩ8 (q).A4 D2(q2+1)/(2,q−1).2 .2 .3 q odd + 1+8 3 PΩ8 (2) 2+ :S3 + 11 1+4 PΩ8 (2).hτi [2 ]:S3.3, 3+ :2S4 + 1+8 3 PΩ8 (3) 3+ :2.A4.2, 2.((A4 o 2) o 2) + 4 13 PΩ8 (3).hτi [2 .3 ] + 6 13 PΩ8 (3).A4 [2 .3 ] + 2 9 2 PΩ8 (3).2 [2 ].S4.2 + 9 PΩ8 (3).4 [2 ].S4.4 + 2 PΩ8 (3).A4 10 :4A4 + 2 9 2 PΩ8 (5).2 [2 ].S4.2 + 9 PΩ8 (5).4 [2 ].S4.4 + 2 + 3 4 2 + 3 4 PΩ12(3) 2 .PΩ4 (3) .2 .S3, 2 .PΩ4 (3) .2 .S3 + 3 + 4 6 PΩ16(3) 2 .PΩ4 (3) .2 .S4

64 4.8. Tables

TABLE VII. Exceptional Lie groups where µ is a graph automorphism deﬁned in Remark (4) on Page 24.

G0 H0 conditions 2 √ 2m+1 Sz(q) [q ]:Zq−1, D2(q−1), Zq± 2q+1:Z4 q = 2 3 √ 2m+1 Ree(q) [q ]:Zq−1, Zq+ 3q+1.Z6 q = 3 ≥ 27 √ Zq+1.Z6, Zq− 3q+1.Z6 0 3 Ree(3) 2 :7, D14, D18 6 2 2 m G2(q).hµi [q ]:Zq−1.hµi, Z(q±1):D12.hµi, Zq2±q+1.Z6.hµi q = 3 ≥ 9 0 1+2 2 G2(2) 3+ :8, 4.S4, 4 :S3 6 G2(3).hµi [3 ]:D8 1+2 2 1+2 2 G2(3) (3+ × 3 ):2S4, (3+ × 3 ):2S4, (SL2(3) ◦ SL2(3)).2 3 2 D4(q) Zq2±q+1:SL2(3), Zq4−q2+1:Z4 q ≥ 2 3 11 1+2 D4(2) [2 ]:(7 × S3), 3+ .2S4 3 11 D4(3) [3 ]:(13 × SL2(3)).2 2 2 F4(q) Zq+1:GL2(3) q ≥ 8 2m+1 2 √ q = 2 Zq± 2q+1.(4 ◦ GL2(3)) q ≥ 32 √ √ . q ≥ 8 Zq2± 2q3+q± 2q+1 Z12 2 0 9 10 2 F4(2) [2 ]:5:4, [2 ]:S3, 5 :4A4 2 F4(2) 13:12, SU3(2):2 2 2 F4(8) 13 :(4 ◦ GL2(3)) 4 1+4 2 F4(q).hµi Zq±1.(2+ :S3).hµi q ≥ 8 m 2 q = 2 Zq2±q+1.(3 × SL2(3)).hµi q ≥ 4 2 Zq2+1.(4 ◦ GL2(3)).hµi q ≥ 4 Zq4−q2+1.Z12.hµi q ≥ 4 2 F4(2) 3.SU3(2) .3.2 22 2 2 F4(2).hµi [2 ]:S3.hµi, 7 :(3 × SL2(3)).hµi 4 1+4 2 F4(4).hµi [5 ].(2+ :S3).hµi 2 3 1+2 E6(q) Zq2−q+1.3 .SL2(3) q ≥ 3 2 3 2 E6(2) 3.SU3(2) .3 .S3 3 1+2 E6(q) Zq2+q+1.3 .SL2(3) E8(q) Zq8±q7∓q5−q4∓q3±q+1.Z30 2 Zq4−q2+1.(Z12 ◦ GL2(3)) 2 4 2 E8(2) 3 .SU3(2) .3 .2S4

65 Chapter 5

Finite edge-primitive s-arc-transitive graphs with s ≥ 4

It turns out that Theorem 4.1.1 has an extensive application in the study of algebraic graph theory. In this short chapter we present the ﬁrst application of Theorem 4.1.1, which is on the study of highly symmetrical graphs.

5.1 The classiﬁcation

In 1973 Weiss determined all edge-primitive graphs of valency three in [130], and then in 1999 he proposed to classify graphs which are edge-primitive and s-arc-transitive with s ≥ 4, motivated by the classiﬁcation given in [43] of graphs which are vertex-primitive or vertex-biprimitive and s-arc-transitive with s ≥ 4. Attempting to solve the problem of Weiss led to a systematic study of edge-primitive graphs in [44]. The class of s-arc- transitive graphs consists of graphs with high degree of symmetry. As mentioned before, the study of this class of graphs was initiated by Tutte’s work on cubic graphs. In the 1980’s, Weiss proved that s-arc-transitive graphs of valency at least 3 are not

8-arc-transitive. This was achieved by determining the amalgam (Gα,G{α,β}) through analysing the local actions of Gα, known as the local analysis. The problem of char- acterising s-arc-transitive graphs for larger values of s thus naturally arises. For this purpose, Cheryl Praeger in the 1990’s initiated an analysis of the action of G on the vertex set, known as the global analysis. The combination of local analysis and global analysis provides a method for studying s-arc-transitive graphs, which is also the main tool for studying the graphs appearing in this thesis. Combining local analysis and global analysis with the outcome of primitive groups with soluble stabilizers, we obtain a complete classiﬁcation of edge-primitive s-transitive graphs with s ≥ 4, given as follows.

66 5.2. A reduction for the proof of Theorem 5.1.1

Theorem 5.1.1. Let Γ be a graph of valency k ≥ 3 which is edge-primitive and s- transitive with s ≥ 4, and let G = AutΓ . Then for an edge {α, β}, (G, Gα,G{α,β}, k) lies in TABLE B.

TABLE B

Aut(Γ ) = G Gα G{α,β} k s remarks 5 PΓL2(9) S4 × 2 [2 ] 3 5 Tutte 8−cage PSL2(17) S4 D16 3 4 Biggs−Smith 2 1+2 5 Ru 5 :GL2(5) 5+ :[2 ] 6 4 Ru graph 2 1+2 M12.2 Z3:(2S4) 3+ :D8 4 4 Weiss 4 6 2 J3.2 [2 ]:(3 × A5).2 [2 ]:S3 5 4 Weiss 2 3 PSL3(q).O.2 [q ]:(Z q−1 .PGL2(q)).O ([q ]:(Zq−1 × Z q−1 )).O.2 q + 1 4 (q−1,3) (q−1,3) 3 4 2 PSp4(q).O.2 [q ]:GL2(q).O ([q ]:Zq−1).O.2 q + 1 5 ( q = 2m) 5 6 2 G2(q).O.2 [q ]:GL2(q).O ([q ]:Zq−1).O.2 q + 1 7 ( q = 3m)

We remark that the graphs appearing in Theorem 5.1.1 are all well-known.

5.2 A reduction for the proof of Theorem 5.1.1

The proof of Theorem 5.1.1 is based on the solubility of edge-stabilizers of s-arc-transitive graphs, and the classiﬁcation of primitive permutation groups with soluble stabilizers in Theorem 4.1.1. Here we reduce the proof of Theorem 5.1.1 to the case of almost simple groups.

5.2.1 s-Arc transitive graphs

Let Γ = (V,E) be a ﬁnite simple connected undirected graph. Recall that An s-arc

of Γ is an ordered sequence of vertices (α0, α1, . . . , αs) such that {αi−1, αi} ∈ E and

αi−1 6= αi+1 for all admissible values of i. A regular graph Γ is called (G, s)-arc-transitive if G ≤ AutΓ is transitive on the set of s-arcs. For a vertex α, denote by Γ (α) the set of Γ(α) vertices to which α is adjacent. Then Gα induces a permutation group Gα on Γ (α), [1] Γ(α) ∼ [1] [1] [1] [1] with kernel Gα . Then Gα = Gα/Gα . For an arc (α, β), let Gαβ = Gα ∩ Gβ , the pointwise stabiliser of Γ (α) ∪ Γ (β).

Example 5.2.1. Let V be a 3-dimensional vector space over the ﬁeld Fq. Let P be the set of 1-subspaces, and let L the set of 2-subspaces of V . Then (P, L) forms a projective plane of order q. Let Γ be the incident graph of (P, L), that is, the vertex

set is P ∪ L, and adjacency is deﬁned by inclusion. Let G = PSL3(q).O.hτi, where

67 5.2. A reduction for the proof of Theorem 5.1.1

O ≤ PΓL3(q)/PSL3(q) and τ is the transpose-inverse. Then G ≤ AutΓ is transitive on vertices and arcs of Γ . Let {α, β} be an edge. Then the stabilisers

2 3 Gα = [q ]:(Z q−1 .PGL2(q)).O, Gαβ = [q ]:(Zq−1 × Z q−1 ).O. (q−1,3) (q−1,3)

Γ(α) ∼ Γ(α) ∼ [1] ∼ 2 Moreover, Gα = PGL2(q).O, and Gαβ = [q]:Zq−1.O, with Gα = [q ]:Z q−1 . Since (q−1,3) [1] 1 6= Gβ ¡ Gαβ, we have

[1] Γ(α) Γ(α) ∼ 1 6= (Gβ ) ¡ Gαβ = [q]:Zq−1.O.

[1] [1] [1] Thus, Gβ is not faithful on Γ (α), and 1 6= Gαβ ¡ Gα . Assume that γ ∈ Γ (α). Then [1] Γ(γ) ∼ it follows that (Gα ) = [q]:Z q−1 . It further implies that if γ ∈ Γ (α) \{β}, then (q−1,3) [1] Γ(γ) [1] Γ(γ) (Gαβ) £ [q]. Hence (Gαβ) is transitive, and so G is 4-arc-transitive on Γ .

Actually, the above example displays the amalgams for 4-transitive graphs. More- over, for any (G, s)-transitive graph Γ with s ≥ 4 and an arc (α, β) of Γ , we have the following conclusion.

2 3 (1) For s = 4, then Gα = [q ]:(Z q−1 .PGL2(q)).O, and Gαβ = [q ]:(Zq−1×Z q−1 ).O, (q−1,3) (q−1,3) f where q = p with p prime, and O . Z(3,q−1) × Zf .

f 3 4 2 (2) For s = 5, then q = 2 , Gα = [q ]:GL2(q).O, and Gαβ = [q ]:Zq−1.O, where O . Zf .

f 5 6 2 (3) For s = 7, then q = 3 , Gα = [q ]:GL2(q).O, and Gαβ = [q ]:Zq−1.O, where O . Zf .

The following brilliant result tells us that these are all the amalgams for s-arc- transitive graphs with s ≥ 4.

Theorem 5.2.2. (Weiss [129]) Let Γ be a (G, s)-transitive graph of valency k ≥ 3, where s ≥ 4. Then s = 4, 5 or 7, k = pf + 1 with p prime, and for an arc (α, β), the stabilizers Gα and Gαβ satisfy the above property (1), (2) or (3). In particular, edge stabilizers of s-arc-transitive graphs with s ≥ 4 are soluble.

Therefore, we have a generic construction of s-arc-transitive graphs, where s ≥ 4. For a group G, let H < G be core-free, and g ∈ G such that g2 ∈ H, and hH, gi = G. Deﬁne a graph with vertex set [G : H] = {Hx | x ∈ G} such that Hx, Hy are adjacent if and only if yx−1 ∈ HgH. Then this graph is exactly the coset graph Cos(G, H, HgH) which we deﬁned in Chapter 3. It is readily seen that this graph is connected and G-arc-transitive.

Lemma 5.2.3. Let G be a group, H < G be core-free.

68 5.2. A reduction for the proof of Theorem 5.1.1

(1) Assume that there exists an element g ∈ G such that g2 ∈ H, hH, gi = G, and g ∼ (H,H ∩ H ) = (Gα,Gαβ), where Gα and Gαβ satisfy condition (1), (2) or (3). Then Γ = Cos(G, H, HgH) is a connected (G, s)-transitive graph, where s = 4, 5 or 7, respectively.

(2) Assume that E < G is such that E ∩ H has index 2 in E, hE,Hi = G, and

(H,H ∩ E) = (Gα,Gαβ), where Gα and Gαβ satisfy the condition (1), (2) or (3). Then each 2-element g ∈ E \ H is such that Γ = Cos(G, H, HgH) is a (G, s)- transitive graph, where s = 4, 5 or 7, respectively.

Proof. (1) For the graph Γ = Cos(G, H, HgH), choose the vertex α = H. Then α is g adjacent to β := Hg. Further, Gα = H, Gαβ = H ∩ H . Now the result follows from Theorem 5.2.2. (2) Let A := H ∩ E < E. Since |E : A| = 2, there exists a 2-element g ∈ E \ A. Thus E = hA, gi < hH, gi. Since hE,Hi = G, we conclude that hH, gi = G. Thus the coset graph Γ = Cos(G, H, HgH) is connected and G-arc-transitive.

We make the following observations on the edge-stabilizers of s-arc-transitive graphs with s ≥ 4, which are used in the proof of Theorem 5.1.1.

Corollary 5.2.4. Let Γ be a connected (G, s)-arc-transitive graph with s ≥ 4. Then the following statements hold, where {α, β} is an edge:

3 4 6 f (1) The Fitting subgroup P of G{α,β} is a p-group of order q , q or q , where q = p with p prime.

(2) The order of G{α,β}/P is divisible by q − 1, and G{α,β}/P has an abelian normal subgroup Q which is isomorphic to Zq−1 × Z q−1 . (q−1,3)

(3) Each prime divisor of |G{α,β}/P | divides q − 1 or 2f; in particular, if q = p is a

prime, then each prime divisor of |G{α,β}| divides q − 1 or equals 2.

(4) If q = p = 2, then G{α,β} is a 2-group of order 16 or 32; if q = p = 3, then if G{α,β} = [3 ]:(Z3f −1 × Z3f −1):Ze.Z2, where i = 3, 4 or 6, and e | f.

5.2.2 Edge-primitive graphs

Let Γ = (V,E) be an edge-primitive graph. Then for an edge {α, β}, the edge stabilizer

G{α,β} is a maximal subgroup of G. Obviously, G acts faithfully on E. Hence, G is a

primitive permutation group on E, with point stabilizer G{α,β}. To avoid the trivial case, we assume that Γ is regular and has valency k ≥ 3. By

[42], Γ is connected and G-arc-transitive. Thus, the arc stabilizer Gαβ is a subgroup of

G{α,β} of index 2, and as Γ has valency k, Gαβ is a subgroup of the vertex stabilizer Gα of index k.

69 5.3. Proof of Theorem 5.1.1

Assume further that Γ is (G, s)-arc-transitive, where s ≥ 4. Then by Theorem 5.2.2, the stabilizer G{α,β} is soluble.

Lemma 5.2.5. The group G is a primitive permutation group on the edge set E with soluble stabilizer G{α,β}.

By Theorem 4.1.1, G is aﬃne, almost simple, or in product action. So (G, G{α,β}) is a pair which lies in TABLES I-VII, and G{α,β} satisﬁes Corollary 5.2.4. In the rest of this section, we show that G is almost simple.

Lemma 5.2.6. The group G is almost simple and has a normal subgroup G0 such that the pair (G0,H0), where H0 = G0 ∩ G{α,β} is a soluble maximal subgroup of G0, lies in TABLES I-VII, and satisﬁes the conditions of Corollary 5.2.4.

Proof. Let N := soc(G). Then N is transitive on E, and so it has at most two orbits on V . Suppose that N is abelian. Then N is regular on E, and semiregular on V . Hence |E| = |N| divides |V |. However, since Γ is of valency q+1 ≥ 3, we have 2|E| = (q+1)|V |, which is not possible. Thus, N is nonabelian. Suppose that N = T l, where T is simple and l ≥ 2. Then G acting on E is in product action, and Γ is N-edge transitive. It follows that N{α,β} has order greater Γ(α) Γ(α) than 2. Thus 1 6= Nα ¡ Gα, and Nα £ PSL2(q). So Nα is 2-transitive, and Γ is locally (N, 2)-arc-transitive. It follows that a normal direct factor T of N is semiregular on V [44], which is not possible. Thus, G is almost simple, as claimed in the lemma.

5.3 Proof of Theorem 5.1.1

Let Γ be a connected G-edge-primitive (G, s)-arc-transitive graph with s ≥ 4. Then by

Lemma 5.2.6, G is almost simple, and there exists a normal subgroup G0 ¡ G such that

(G, G{α,β}) is one of the pairs (G0,H0) given in TABLES I-VII. Further, let P = Op(H0), f where p is the prime such that q = p appeared in the amalgam (Gα,G{α,β}).

Case 1. Assume that G0 = An or Sn. By Lemma 4.2.1, either n = p is a prime and ∼ P = Zp, or n ≤ 16 and p = 2 or 3, and the pair (G0,H0) lies in TABLE I. Inspecting the candidates in TABLE I, by Lemma 5.2.4, we conclude that n = 5, 6 or 9. It is easily shown that the only possibility is n = 6, which corresponds to the Tutte’s 8-cage. This graph is a (G0, 4)-arc-transitive cubic graph, for G0 = PGL(2, 9) or M10, and has full automorphism group PΓL(2, 9).

Case 2. Assume that G0 is a sporadic almost simple group. Then (G0,H0) lies in TABLE II.

70 5.3. Proof of Theorem 5.1.1

3 1+2 Suppose that p ≥ 7. Since |P | ≥ p , an inspection of TABLE II shows that P = p+ .

Thus f = 1, and the arc stabiliser Gαβ should be the abelian subgroup of H0/P of index 1+2 2. However, the only possibilities for P = p+ are (G0, p) = (J4, 11), (Fi24, 7), (He, 7), 0 (O N.2, 7) or (M, 13), none of which is such that H0/P has an abelian subgroup of index 2. This is a contradiction. 1+2 1+4 Suppose that p = 5. Inspecting TABLE II, we obtain that P = 5+ or 5+ ,

corresponding to the cases where G0 = Ru, Co2, Th, HS.2, McL, or HN. Hence ∼ 2 q = p = 5, the vertex stabiliser Gα = 5 :GL(2, 5), and H0 has an index 2 sub- ∼ 1+2 1+2 5 group Gαβ = 5+ :(Z4 × Z4). Thus, we further have that (G0,H0) = (Ru, 5+ :[2 ]) 1+2 5 or (HS.2, 5+ :[2 ]). By the Atlas [19], there is no subgroup of HS.2 which is isomor- 2 phic to 5 :GL(2, 5). We conclude that G0 = Ru. In this case A := NG0 (P ) is an

index two subgroup of H0, and A is contained in a maximal subgroup of G0 which 2 is isomorphic to 5 :GL(2, 5). Thus by Lemma 5.2.3, there exists a G0-edge-primitive

(G0, 4)-arc-transitive graph Γ . This graph is known as the Ru-graph, which is also vertex-primitive. It was ﬁrst constructed by Stroth and Weiss in [109]. Suppose that p = 3. All the possible candidates in TABLE II are such that |P | ≤ 311, and thus f ≤ 3. Further, we observe the following

3 6 (i) If f = 1, then Gαβ = [3 ]:(Z2 × Z2) or [3 ]:(Z2 × Z2).

3 6 (ii) For f = 2, then Gαβ = [9 ]:(Z8 × Z8).O or [9 ]:(Z8 × Z8).O, where O = 1 or 2.

3 6 (iii) For f = 3, then Gαβ = [27 ]:(Z26 × Z26) or [27 ]:(Z26 × Z26). In particular,

H0 = G{α,β} has order divisible by 13.

Inspecting TABLE II leads to the only possibilities: either G0 = M12.2 and H0 = 1+2 1+8 1+6 3+ :D8, or G0 = Fi23 and H0 = 3+ :2− . The latter is not possible since H0 = 1+8 1+6 3+ :2− has no elements of order 8. Therefore, G0 = M12.2. By the Atlas [19], there ∼ 2 3 2 exists a maximal subgroup M9:S3 = 3 :2S4 which intersects H0 at [3 ]:Z2. By Lemma

5.2.3, there exists a G0-edge-primitive (G0, 4)-arc-transitive graph Γ . This graph was ﬁrst constructed by Weiss [126].

Finally, suppose that p = 2. If q = p = 2, then H0 = D8 or D8 × Z2; however,

an inspection of candidates of Table II shows that no such H0 exists. Thus, f ≥ 2. 3 3 f Suppose that f ≥ 3. Then O2(H0) ≥ (2 ) , and 2 − 1 divides |H0|; however, none 3 of the candidates in TABLE II has this property. Hence, f = 2, and H0 = [4 ]:Z3.O, 4 or [4 ]:(Z3 × Z3).O, where O = 1 or 3. The only candidate in TABLE II with H0 6 having this form is G0 = J3 and H0 = [2 ]:(3 × S3). By the Atlas [19], there exists a 4 subgroup of G0 = J3 which is isomorphic to [2 ]:GL2(4). Weiss [127] proved that there

is a (G0, 4)-arc-transitive graph Γ corresponding to this case.

71 5.3. Proof of Theorem 5.1.1

Thus for sporadic almost simple groups, we have found three edge-primitive s- transitive-graphs with s ≥ 4, as listed in TABLE B.

Case 3. Assume that G is a classical group of Lie type. Then the pair (G0,H0) lies ∼ in TABLES III-VI. Since PSL(2, 9) = A6, we assume that soc(G) 6= PSL(2, 9). Suppose ﬁrst that q = 2f with f ≥ 1. Then, by Theorem 5.2.2, either

4 5 (i) q = 2, Γ is cubic, and G{α,β} = [2 ] or [2 ], or

3f f 4f (ii) Gα = [2 ]:GL2(2 ).O, and Gαβ = [2 ]:(Z2f −1 × Z2f −1).O, where O ≤ Zf .

For the case (i), Γ is cubic, and an inspection shows that the candidates for G0 in

TABLES III-VI with H0 = G{α,β} of order 16 or 32 are G0 = PGL2(7), PSL2(17) or

PSL2(31). Further, by Theorem 5.2.2, the vertex stabiliser Gα is isomorphic to S4 or

S4 × S2, respectively. By the Atlas [19], the group PSL2(31) does not have a subgroup which is isomorphic to S4 or S4 × S2, and thus G0 6= PSL2(31). For the remaining cases, each case corresponds to a known graph:

(a) The case (G0,H0) = (PGL2(7), D16) corresponds to the Heawood graph, which is 4-transitive;

(b) The case (G0,H0) = (PSL2(17), D16) corresponds to the Biggs-Smith graph, which is 4-transitive.

Next we consider the case (ii). Inspecting TABLES III-VI we conclude that (G0,H0) = 4 2 (PSp4(q).hτi, [q ]:Zq−1.2), where q is even and τ is a graph automorphism of PSp4(q). This case corresponds to a family of 5-transitive graphs (see for instance [43]). All these graphs are included in TABLE B. Suppose now that q is odd. By Theorem 5.2.2, the edge stabiliser satisﬁes either

3 (i) G{α,β} = [q ]:(Zq−1 × Z q−1 ).O, where O ≤ Z(q−1,3) × Zf , or (q−1,3)

6 (ii) G{α,β} = [q ]:(Zq−1 × Zq−1).O, where O ≤ Zf .

Recall that G{α,β} = H0 is a maximal subgroup of G0 given in TABLES II-VI. Inspecting the group pairs (G0,H0) in these tables, we obtain only one possibility: soc(G) =

PSL3(q). It is well known that this case corresponds to the projective plane graph (see Example 5.2.1), as included in TABLE B.

Case 4. Assume that G is an exceptional group of Lie type. Then the pair (G0,H0) lies in TABLE VII. Recall that the soluble subgroup H0 is the edge stabiliser, given in

Theorem 5.2.2. Inspecting the group pairs (G0,H0) in TABLE VII, we conclude that f the only possibility is G0 = G2(q).hτi, where q = 3 , and τ is a graph automorphism of

72 5.3. Proof of Theorem 5.1.1

G2(q). For this case, it is well known that the corresponding graphs form the family of the classical generalized hexagon graphs all of which are 7-transitive. This completes the proof of Theorem 5.1.1.

73 Chapter 6

Finite 2-path-transitive graphs

Before presenting the second application of Theorem 4.1.1, in this chapter we slightly deviate our theme by undertaking a general study on 2-path-transitive graphs, which is very interesting in its own right.

6.1 The problem and the results

Let Γ = (V,E) be a graph with vertex set V and edge set E, and let (α, β, Γ ) be a 2-arc such that β is adjacent to both α and Γ . Recall that a 2-path [α, β, Γ ] is obtained by identifying the two 2-arcs (α, β, Γ ) and (Γ , β, α). Let G ≤ AutΓ be a group of automorphisms of Γ . Then Γ is called (G, 2)-path-transitive if G acts transitively on the set of 2-paths of Γ .A(G, 2)-path-transitive graph is sometimes simply called a 2-path-transitive graph. This parallels the deﬁnition of 2-arc-transitive graph, that is, a subgroup G ≤ AutΓ is transitive on the set of 2-arcs of Γ . The class of 2-path-transitive graphs is slightly larger than the class of 2-arc-transitive graphs. The following problem therefore naturally arises.

Problem Characterize 2-path-transitive graphs that are not 2-arc-transitive.

For a group G ≤ AutΓ that is 2-path-transitive but not 2-arc-transitive on Γ , the vertex stabiliser has odd order and simple structure. The ﬁrst result of this chapter is the following.

Theorem 6.1.1. Let Γ be a connected (G, 2)-path-transitive graph which is not (G, 2)- arc-transitive. Then Γ is G-arc-transitive and has valency pe, where pe ≡ 3 (mod 4) Γ(α) e with p prime, and for an edge {α, β}, Gα ≤ AΓL(1, p ) is 2-homogeneous, and the following hold:

[1] ∼ [1] Γ(α) Γ(α) e (i) Gβ = (Gβ ) ¡ Gαβ ≤ Z(pe−1)/2:Ze < ΓL(1, p );

74 6.1. The problem and the results

e [1] e [1] [1] [1] ∼ (ii) Gα = Zp:Gαβ = (Gα × (Zp:Gβ )).O, and Gαβ = (Gα × Gβ ).O, where O = Γ(α) [1] Γ(α) Gαβ /(Gβ ) ;

e (pe−1)e 2 (iii) |Gα| is odd and divides p ( 2 ) .

The ﬁrst small values for the valency of a 2-path-transitive but not 2-arc-transitive graphs are 3, 7, 11, 19, 23, 27, and 31. The next theorem determines the amalgam for the full automorphism group of a 2-path-transitive graph.

Theorem 6.1.2. Let Γ be a connected (G, 2)-path-transitive graph of valency pe with p prime. Assume that Γ is not (G, 2)-arc-transitive. Then, for G ≤ X ≤ AutΓ , one of the following statements holds:

e e (i) Xα is soluble, and either Γ is cubic, or Xα ≤ ΓL(1, p ) × AΓL(1, p ).

b b (ii) ASL(a, p ) ≤ Xα ≤ AGL(a, p ).b, or

b b b b SL(a, p ) × ASL(a, p ) ≤ Xα ≤ GL(a, p ).b × AGL(a, p ).b, where ab = e;

e (iii) Xα = An, Sn, An−1 × An, An−1 × Sn or Sn−1 × Sn, where n = p ;

e (iv) Xα = PSL(2, 11) or A5 × PSL(2, 11), and p = 11;

e Xα = M11 or M10 × M11, and p = 11;

e Xα = M23 or M22 × M23, and p = 23;

(v) the valency pe = 7, and either

2 (a) Xα = GL(3, 2), Z2 × GL(3, 2), or A4 × GL(3, 2), or S4 × GL(3, 2), or 3 4 6 (b) Xα = 2 :GL(3, 2), [2 ].GL(3, 2), or M.(L × GL(3, 2)), where M = [2 ] or 20 [2 ], and L = 1, Z3 or S3.

One of the principal motivations of our interest in 2-path-transitive graphs comes from a relationship between this class of graphs and half-transitive graphs, as shown in the following result.

Theorem 6.1.3. A graph is 2-path-transitive but not 2-arc-transitive if and only if its line graph is half-transitive.

We observe that a (G, 2)-path-transitive cubic graph is exactly a G-arc-regular graph, that is, G acts regularly on the set of arcs. Thus, the concept of 2-path-transitive graphs, as a generalisation of 2-arc-transitive graphs, may also be viewed as a generalisation of cubic arc-regular graphs. The special case of Theorem 6.1.3 for cubic graphs was ﬁrst obtained by Maruˇsiˇcand Xu in [87, Proposition 1.1], which motivated an extensive study on 1-arc regular graphs [30, 33]. Theorem 6.1.3 thus provides us with a method for constructing half-transitive graphs. Here is an example.

75 6.2. Proofs of Theorems 6.1.1 and 6.1.2

Theorem 6.1.4. For each q = pe =∼ 3 (mod 4) with p prime and q ≥ 19, there exists a connected 2-path-transitive graph Γ of valency q such that AutΓ = Aq × Zc with c | e, and the line graph L(Γ ) is a half-transitive graph of valency 2(q − 1).

6.2 Proofs of Theorems 6.1.1 and 6.1.2

By deﬁnition, a 2-path-transitive graph is edge-transitive. Furthermore, we have the following simple lemma, the proof of which is easy and omitted.

Lemma 6.2.1. A regular 2-path-transitive graph is arc-transitive.

Recall that a permutation group G on Ω is called 2-homogeneous if G induces a transitive action on the set of 2-subsets of Ω. As mentioned before, a 2-transitive group is 2-homogeneous, but the converse is not necessarily true. Let Γ be a graph and G ≤ AutΓ . For α ∈ V , as before, we use Γ (α) to denote the neighborhood of α. Then each ordered pair of and each 2-subset of vertices in Γ (α) exactly corresponds to a 2-arc and a 2-path, respectively, with α being the centre vertex.

Thus, if Γ is (G, 2)-arc-transitive or (G, 2)-path-transitive, then Gα is 2-transitive or 2-homogeneous on Γ (α), respectively. Hence Γ is G-locally-2-transitive or G-locally-2- homogeneous, respectively.

Conversely, if G is transitive on V , and Gα is 2-transitive or 2-homogeneous on Γ (α) for some vertex α, then Γ is (G, 2)-arc-transitive or (G, 2)-path-transitive. Thus, we have the following statement, which was ﬁrst observed by Conder and Praeger (1996) Γ(α) in [17]. As usual, the permutation group induced by Gα on Γ (α) is denoted by Gα , [1] Γ(α) ∼ [1] and the kernel (of Gα acting on Γ (α)) is denoted by Gα . Then Gα = Gα/Gα .

Lemma 6.2.2. Let Γ be a connected G-vertex-transitive graph. Then Γ is (G, 2)- Γ(α) path-transitive but not (G, 2)-arc-transitive if and only if, for a vertex α, Gα is 2- homogeneous but not 2-transitive.

Finite 2-homogeneous permutation groups which are not 2-transitive have relatively simple structure and were classiﬁed by Kantor as given in the following result.

Proposition 6.2.3. (see [24, Theorem 9.4B]) Suppose that a permutation group P is 2-homogeneous of degree n but not 2-transitive. Then n = pe ≡ 3 (mod 4), where p is prime and e is odd, and further,

pe(pe−1) (a) |P | is odd and divisible by 2 , and

(b) ASL(1, pe) ≤ P ≤ AΓL(1, pe).

76 6.2. Proofs of Theorems 6.1.1 and 6.1.2

[1] [1] [1] Let (α, β) be an arc in Γ . Again we deﬁne Gαβ = Gα ∩ Gβ , the kernel of the

arc stabiliser Gαβ acting on the double star Γ (α) ∪ Γ (β). Recall that a graph Γ is

called G-locally-primitive if Gα acts primitively on Γ (α) for every vertex α. The class of locally-primitive graphs is larger than the class of 2-path-transitive graphs. Then a fundamental result regarding locally-primitive graphs is the well-known Thompson- Wielandt Theorem and Weiss’s theorem.

Theorem 6.2.4. Let Γ be a connected G-locally-primitive arc-transitive graph. Then [1] the double star kernel Gαβ is a p-group for some prime p, and further, the following statements hold:

Γ(α) [1] (1) (Weiss [124]) If Gα is aﬃne and |Γ (α)| ≥ 5, then Gαβ = 1.

(2) (Weiss [124]) Assume that Γ is (G, s)-arc-transitive with s ≥ 2. Then either

[1] (i) Gαβ = 1, and s ≤ 3, or, [1] Γ(α) qd−1 (ii) Gαβ is a non-trivial p-group, and soc(Gα ) = PSL(d, q) with |Γ (α)| = q−1 , d ≥ 2 and q = pf .

Based on this result, the vertex stabilizer Gα is classiﬁed (mainly due to Weiss and Troﬁmov). Here we need a special case.

Γ(α) Corollary 6.2.5 ([118]). Assume that Gα £ GL(3, 2) and |Γ (α)| = 7. Then either

[1] [1] [1] (i) Gαβ = 1, Gα ¡ S4, and Gα = Gα × GL(3, 2), or

[1] 3 4 6 20 (ii) Gαβ is a nontrivial 2-group, and |O2(Gα)| = 2 , 2 , 2 , or 2 .

Now we are ready to prove Theorem 6.1.1.

Proof of Theorem 6.1.1: Assume that Γ is a connected (G, 2)-path-transitive graph. Γ(α) Let α be a vertex of Γ . By Lemma 6.2.2, the permutation group Gα is 2-homogeneous, Γ(α) e and is not 2-transitive. Thus, by Proposition 6.2.3, we have Gα ≤ AΓL(1, p ), and Γ(α) Γ(α) e Γ(α) hence the point stabiliser Gαβ = (Gα )β ≤ ΓL(1, p ). Since Gα is not 2-transitive, Γ(α) it follows that Gαβ ≤ Z(pe−1)/2:Ze. Since Γ is G-arc-transitive, there exists an element g ∈ G which interchanges α [1] ∼ [1] [1] [1] [1] and β. Thus, Gα = Gβ . By Theorem 6.2.4, we have Gα ∩ Gβ = Gαβ = 1. Since [1] Gβ ¡ Gαβ, we have

[1] ∼ [1] [1] [1] ∼ [1] [1] [1] ∼ [1] Γ(α) Gβ = Gβ /(Gα ∩ Gβ ) = (Gα Gβ )/Gα = (Gβ ) ,

as in part (i) of Theorem 6.1.1.

77 6.2. Proofs of Theorems 6.1.1 and 6.1.2

[1] [1] [1] [1] Since Gα and Gβ normalize each other and Gα ∩ Gβ = 1, we conclude that [1] [1] Γ(α) [1] Γ(α) Gα × Gβ ¢ Gαβ. Let O = Gαβ /(Gβ ) . Then

[1] [1] ∼ [1] [1] [1] [1] ∼ Γ(α) [1] Γ(α) Gαβ/(Gα Gβ ) = (Gαβ/Gα )/(Gα Gβ /Gα ) = Gαβ /(Gβ ) = O.

[1] [1] Hence Gαβ = (Gα × Gβ ).O, as in Theorem 6.1.1 (ii). [1] 0 Since Gα is isomorphic to a subgroup of Z(pe−1)/2:Ze, it follows that a Hall p - [1] 0 subgroup of Gα is normal and has index dividing e. Let H be the Hall p -subgroup of [1] Gα . Then C [1] (H) is of order coprime to p. Since H is metacyclic, H has a cyclic Gα normal subgroup N such that H/N is cyclic. Let P be a Sylow p-subgroup of Gα. Then |P | is divisible by pe, and P acts by conjugation on H. Since both N and H/N have order coprime to p, if any element g ∈ P acts nontrivially on H, then x must act nontrivially on N or on H/N. However, since N and H/N are cyclic, |Aut(N)| < |N| and |Aut(H/N)| < |H/N|, and further, as |H| < pe ≤ |P |, some element x ∈ P \{1} acts nontrivially on H. [1] [1] [1] Let C = CGα (H). Then C ∩ P 6= 1, and (C ∩ P ) ∩ Gα = 1. Let C = (CGα )/Gα . [1] Γ(α) ∼ e Then C ¡ Gα/Gα , and C contains elements of order p. Since M := soc(Gα ) = Zp Γ(α) is the only minimal normal subgroup of Gα , we obtain that M ≤ C. It then follows ∼ e that Op(Gα) = M = Zp, as in Theorem 6.1.1 (ii). Γ(α) By Proposition 6.2.3, the 2-homogeneous permutation group Gα is of odd order, [1] ∼ [1] [1] Γ(α) and so is Gα (= Gβ ). Hence Gα is of odd order, and the order divides |Gα ||Gα |, e (pe−1)e 2 which divides p ( 2 ) , as in part (iii) of Theorem 6.1.1.

The proof of Theorem 6.1.2 depends on Theorem 6.2.4 and the classiﬁcation of 2- transitive permutation groups.

Proof of Theorem 6.1.2. Let Γ be a connected (G, 2)-path-transitive graph of valency k. Assume further that Γ is not (G, 2)-arc-transitive. Let G ≤ X ≤ AutΓ . Then

Gα ≤ Xα, and Γ(α) Γ(α) Gα ≤ Xα .

Γ(α) Γ(α) Since Gα is 2-homogeneous, so is Xα . Γ(α) Assume ﬁrst that Xα is soluble. By Tutte’s result, we may assume that Γ is not of valency 3. Then, by Theorem 6.1.1, k = pe ≡ 3 (mod 4), and so k ≥ 7. Thus, the classiﬁcation of 2-transitive soluble groups (Huppert 1957, refer to [78]) tells us that Γ(α) e Γ(α) e Xα ≤ AΓL(1, p ), and so Xαβ ≤ ΓL(1, p ). On the other hand, by Theorem 6.2.4, [1] we have Xαβ = 1, and so

∼ [1] ∼ Γ(α)∪Γ(β) Γ(α) Γ(β) e e Xαβ = Xαβ/Xαβ = Xαβ ≤ Xαβ × Xαβ ≤ ΓL(1, p ) × ΓL(1, p ).

78 6.2. Proofs of Theorems 6.1.1 and 6.1.2

e Since Xα = Zp:Xαβ, it follows that

e e Xα ≤ ΓL(1, p ) × AΓL(1, p ).

Γ(α) Γ(α) We next assume that Xα is insoluble. Then Xα is 2-transitive, and hence Γ(α) Xα is aﬃne, or almost simple. Γ(α) Γ(α) e Γ(α) Γ(α) Let Xα be aﬃne. Then Xα = Zp:Xαβ , where Xαβ ≤ GL(e, p). Since

e Γ(α) ∼ Γ(α) Γ(α) e Γ(α) Zp:Gαβ = Gα < Xα = Zp:Xαβ ,

Γ(α) we have Z(pe−1)/2 < Xαβ ≤ GL(e, p). Noticing that e is odd and H is insoluble, by the classiﬁcation of 2-transitive aﬃne groups (see [78]), we obtain H £ SL(a, pb), Sp(a, qb) 6 Γ(α) or G2(q ), where ab = e. Let Z be the centre of Gαβ . Then GαβZ/Z is a Singer cycle of PΓL(a, qb). It follows from [46, Theorem 1.49] that H £ SL(a, qb). By Theorem 6.2.4, [1] [1] [1] b Xαβ = 1, and it then follows that either Xα ≤ Zpb−1 with b | e, or Xα £ SL(a, p ), as in part (ii). Γ(α) Suppose now that Xα is an almost simple 2-transitive permutation group. Since |Γ (α)| = pe is a prime-power, by [47], one of the following holds:

Γ(α) e (a) Xα = An or Sn, with n = p ≥ 7;

Γ(α) e (b)( Xα , p ) = (PSL(2, 11), 11), (M11, 11), or (M23, 23);

Γ(α) e qd−1 f (c) Xα £ PSL(d, q), and p = q−1 where q = r with r prime. [1] [1] ∼ For the cases (a) and (b), by Theorem 6.2.4, the stabiliser Xαβ = 1, and so Xα = [1] Γ(β) Γ(β) Γ(β) [1] [1] Γ(β) (Xα ) ¡Xαβ . Since Xαβ is almost simple, either Xα = 1, or Xα ≥ £soc(Xαβ ).

Thus, Gα is as in part (iii) or part (iv) of Theorem 6.1.2. qd−1 rdf −1 e qd−1 For case (c), q−1 = rf −1 = p . A subgroup of PΓL(d, q) of order q−1 is isomorphic d e to a subgroup of ΓL(1, q )/Zq−1. It follows that Zp is cyclic, that is, e = 1, and hence d e Gα = Zp:Z(p−1)/2 ≤ ΓL(1, q )/Zq−1. It then is easily shown that d = 3, and p = 7. [1] [1] ∼ [1] Γ(β) Γ(β) Suppose that Xαβ = 1. Then Xα = (Xα ) ¡Xαβ = S4. Hence Xα = GL(3, 2), 2 or 2 × GL(3, 2), or A4 × GL(3, 2), or S4 × GL(3, 2). [1] [1] Suppose that Xαβ 6= 1. By Theorem 6.2.5, we have that Xαβ is a 2-group, and 3 4 6 20 [1] ∼ |O2(Xα)| = 2 , 2 , 2 , or 2 . Since Xα/Xα = GL(3, 2) is almost simple, it follows that [1] 3 4 3 4 O2(Xα) = O2(Xα ). If |O2(Xα)| = 2 or 2 , then Xα = 2 .GL(3, 2), or [2 ].GL(3, 2). If 6 20 |O2(Xα)| = 2 or 2 , then

[1] Xα = O2(Xα ).(L × GL(3, 2)), where L = 1, Z3 or S3, as in part (v).

79 6.3. Line graphs

6.3 Line graphs

Here we study line graphs of edge-transitive graphs, and complete the proof of Theo- rem 6.1.3.

Lemma 6.3.1. Let Γ be an undirected connected graph, and G ≤ AutΓ . Then the following hold:

(i) Γ is (G, 2)-path-transitive if and only if L(Γ ) is G-edge-transitive;

(ii) Γ is (G, 2)-arc-transitive if and only if L(Γ ) is G-arc-transitive;

(iii) Γ is (G, 2)-path-transitive and is not (G, 2)-arc-transitive if and only if L(Γ ) is G-half-transitive.

Proof. (i) Assume that Γ is (G, 2)-path-transitive. Let {u1, u2}, {v1, v2} be two edges of L(Γ ), where ui, vi are edges of Γ . Since u1 is incident with u2, and v1 is incident with v2, we can assume that u1 = {α, β}, u2 = {β, γ} and v1 = {α1, β1}, v2 = {β1, γ1}. Thus

[α, β, γ] and [α1, β1, γ1] are two 2-paths of Γ . Since Γ is (G, 2)-path-transitive, there g g exists g ∈ G such that [α, β, γ] = [α1, β1, γ1], from which we conclude that β = β1, and g g {α, γ} = {α1, γ1}. Therefore, {u1, u2} = {v1, v2}, that is, L(Γ ) is G-edge-transitive.

Conversely, assume that L(Γ ) is G-edge-transitive. Let [α, β, γ] and [α1, β1, γ1] be t- wo 2-paths of Γ . Then {{α, β}, {β, Γ }} is an edge of L(Γ ), and {{α1, β1}, {β1, Γ1}} is an- g other edge of L(Γ ). So there exists g ∈ G such that {{α, β}, {β, Γ }} = {{α1, β1}, {β1, Γ1}}, g g which implies that β = β1, and [α, β, γ] = [α1, β1, γ1]. Therefore, Γ is (G, 2)-path- transitive.

(ii) Assume that Γ is (G, 2)-arc-transitive. Let (u1, u2), (v1, v2) be two arcs of

L(Γ ). Then u1 = {α, β}, u2 = {β, γ} and v1 = {α1, β1}, v2 = {β1, γ1} for vertices α, β, Γ , α1, β1, Γ1. Since Γ is (G, 2)-arc-transitive, there exists g ∈ G such that g g (α, β, Γ ) = (α1, β1, Γ1). It follows that (u1, u2) = (v1, v2), that is, L(Γ ) is G-arc- transitive.

Conversely, assume that L(Γ ) is G-arc-transitive. Let (α, β, γ) and (α1, β1, γ1) be two 2-arcs of Γ . Suppose that an element g ∈ G sends the arc ({α, β}, {β, Γ }) of L(Γ ) g g to the arc ({α1, β1}, {β1, Γ1}). Then {α, β} = {α1, β1}, and {β, Γ } = {β1, Γ1}. Thus g g g g β = β1, α = α1, and Γ = Γ1, that is, (α, β, γ) = (α1, β1, γ1). Therefore, Γ is (G, 2)-arc-transitive. (iii) Assume that Γ is (G, 2)-path-transitive but not (G, 2)-arc-transitive. Then by Lemma 6.2.1, Γ is G-arc-transitive, which implies that L(Γ ) is G-vertex-transitive. By (ii) and (iii), L(Γ ) is G-edge-transitive but not G-arc-transitive, that is, L(Γ ) is G-half- transitive. The converse is clearly true.

In order to prove Theorem 6.1.3, we need a lemma of Whitney [3, page 1455].

80 6.4. Proof of Theorem 6.1.4

Lemma 6.3.2. (Whitney 1932) If a graph Γ has at least 5 vertices, then AutΓ =∼ Aut(L(Γ )).

Proof of Theorem 6.1.3: Let Γ = (V,E) be a connected 2-path-transitive graph which is not 2-arc-transitive. Then Γ is of valency k ≥ 3, and hence the order |V | ≥ 4.

If |V | = 4, then Γ = K4, which is 2-arc-transitive, a contradiction. Thus, |V | > 4, and by Lemma 6.3.2, we have that AutΓ =∼ AutL(Γ ). By Lemma 6.3.1 (iii), the line graph L(Γ ) is half-transitive. Conversely, let Σ = L(Γ ) be a half-transitive graph of valency m. Since Σ is not arc-transitive, the valency m ≥ 3. Thus, Γ is of valency k ≥ 3, and m = 2(k − 1) ≥ 4.

Suppose that Γ has at most 4 vertices. It follows that Γ = K4. Thus, Σ = L(K4) =

K2,2,2, a complete 3-partite graph of 6 vertices, and so Σ is arc-transitive, which is a contradiction. Hence Γ has at least 5 vertices. By Lemma 6.3.2, the automorphism group AutΓ is isomorphic to AutL(Γ ). By Lemma 6.3.1 (iii), the original graph Γ is 2-path-transitive and is not 2-arc-transitive.

6.4 Proof of Theorem 6.1.4

Let Γ be a connected G-arc-transitive graph and let (α, β) be an arc of Γ . Then there exists a 2-element g ∈ G such that αg = β and βg = α. It follows that g interchanges 2 Gα and Gβ, and hence g normalizes Gαβ and g ∈ Gαβ. Since Γ is connected, it

follows that hH, gi = G. As Gα is transitive on Γ (α), we may write Γ (α) = {Gαgh |

h ∈ Gα}. As indicated in Chapter 3, the graph Γ can be presented as a coset graph

Cos(G, Gα,GαgGα). The following is a direct result of Lemma 6.2.1.

Lemma 6.4.1. Let G be a ﬁnite group with a core-free subgroup H and a 2-element g. Assume that g2 ∈ H, and hH, gi = G,

and the coset action of H on [H : H ∩ Hg] is 2-homogeneous but not 2-transitive. Then the coset graph Cos(G, H, HgH) is (G, 2)-path-transitive but not (G, 2)-arc-transitive.

Let Γ = Cos(G, H, HgH), and let Aut(G, H) = {σ ∈ Aut(G) | Hσ = H}. By Lemma 7.3.6, the graph Γ = Cos(G, H, HgH) is isomorphic to Σ = Cos(G, H, HgσH). Moreover, σ induces an automorphism of Γ if and only if HgH = HgσH. Now we construct some 2-path-transitive graphs.

e Construction 6.4.2. Let q = p ≡ 3 (mod 4) with p prime and e odd. Let T = Aq e naturally act on the set Π = {1, 2, . . . , q}. Let H = K:L < T , where K = Zp, and

81 6.4. Proof of Theorem 6.1.4

L = Z(q−1)/2. We choose an involution g ∈ NT (L)\L, and we deﬁne a coset graph Γ = Cos(T, H, HgH). Since subgroups isomorphic to L are conjugate, we can take L = hti, where

t = (1, 3, . . . , q − 2)(2, 4, . . . , q − 1), and g can be chosen as follows:

q−1 q+3 q+1 q+5 g = (3, q − 2)(5, q − 4) ... ( 2 , 2 )(4, q − 1)(6, q − 3) ... ( 2 , 2 ) ∈ Aq.

Lemma 6.4.3. Let Γ be the graph constructed in Construction 6.4.2. If q ≥ 19, then Γ is a connected graph of valency q.

Proof. Let X = hH, gi.

Since H < Aq and g ∈ Aq, we conclude that X ≤ Aq. As H is 2-homogeneous on Π, so is X. Thus X is aﬃne, or almost simple. e e Suppose that X is aﬃne. Then Zp:Z(pe−1)/2 = H < X ≤ AGL(e, p) = Zp:GL(e, p), e and Dpe−1 = hL, gi < X = AGL(e, p). Since p − 1 is coprime to p, we have that ∼ e Xα = GL(e, p) has a subgroup D = Dpe−1. Now p − 1 has a primitive prime divisor r

e e say. Let z be an element of D of order r. Then Dp −1 = D ≤ NXα (hzi) ≤ ΓL(1, p ). Since e is odd, this is not possible. Thus, the group X is almost simple with socle S. Then X is 2-transitive on Π.

Since the degree q is a prime-power and q ≥ 19, by [47], we have S = Aq, PSL(d, r) with rd−1 q = r−1 , or M23 with q = 23. e rd−1 Suppose that S = PSL(d, r). Then Zp:Z(pe−1)/2 = H < PΓL(d, r) and q = r−1 .A rd−1 subgroup of PΓL(d, r) of order r−1 is a Singer cycle. It follows that e = 1 and q = p, e and hence Zp:Z(p−1)/2 = H ≤ PΓL(d, r), which is not possible since p ≥ 19.

Suppose now that q = 23 and S = M23. Then by the Atlas [19], we have that

NS(L) = Z11:Z5, and so g∈ / X, which is not possible. Therefore, S = Aq, and Γ is connected.

Next we determine the full automorphism group of Γ .

Lemma 6.4.4. The centralizer CAutΓ (T ) is a cyclic group of order dividing e.

Proof. Let C = CAutΓ (T ), and let Y = TC = T × C. Let Y = Y/C, and let α be the vertex of ΓC which is the image of α. Then we have the following relation

∼ Tα ¡ Yα = Y α < Y = T,

82 6.4. Proof of Theorem 6.1.4

and thus Yα is isomorphic to a subgroup of NT (Tα).

Since Tα < T is a 2-homogeneous group of degree q, so is NT (Tα). Since H = Tα is e normal in NT (Tα), the 2-homogeneous group NT (Tα) is aﬃne, and so NT (Tα) = Zp:Nω, where Nω ≤ GL(e, p). It then follows that Z(pe−1)/2 = L ¡ Nω = NT (L), Yα . Nω =

ΓL(1, q), and NT (Tα) = AΓL(1, q). ∼ ∼ On the other hand, C = Y/T = Yα/Tα. Since Yα . NT (Tα) = AΓL(1, q) and e Tα = Zp:Z(pe−1)/2, we conclude that C . Z2 × Ze. Since Γ is (Y, 2)-path-transitive (therefore Γ is locally primitive) and C has at least three orbits on V , it follows that ∼ ∼ the quotient ΓC is (Y, 2)-path-transitive, where Y = Y/C = T = Aq. Suppose that ∼ |C| is even. Then C = Z2 × Zc, where c | e. Thus, Y α β = Zpe−1:Zc, where β ∈ ΓC (α). e e However, since (p − 1)/2 is odd, there is no element of order p − 1 in Ape , which is

a contradiction. Thus, C is of odd order, and so |C| divides e and Y = T × Zc with c | e.

Recall that the soluble radical of a group is the largest soluble normal subgroup.

Lemma 6.4.5. Let R be the soluble radical of AutΓ . Then R ≤ Ze, and RT = R × T .

Proof. Assume that R 6= 1, and let Y = R:T . Let N be a minimal characteristic l subgroup of R. Then N is abelian, say N = Zr with l ≥ 1 and r prime, and the l e e subgroup Z := N:T is such that |Zα| = |N||Tα| = r p (p −1)/2. Since Z is an overgroup of T , Zα satisﬁes Theorem 6.1.2. It follows that Zα satisﬁes part (i) of Theorem 6.1.2. l e Thus, r divides (p − 1)e. Suppose that T does not centralise N. Then T = Aq has m an irreducible representation over Zr with m ≤ l. Since q ≥ 19, by [62, Prop 5.3.7], e l e l p − 2 = q − 2 ≤ m ≤ l. This contradicts r | (p − 1)e. So T centralises N = Zr, and by Lemma 6.4.4, N = Zr and Z = T × Zr. Let C be the maximal characteristic subgroup of R which is centralized by T . Then ∼ C 6= 1, and by Lemma 6.4.4, C = Zc with c | e. Suppose that C 6= R. Then Y = Y/C =

R:T acts 2-path-transitively on ΓC . Arguing as in the previous paragraph for Y in the ∼ place of Y , we conclude that T centralizes a normal subgroup D of R, where D = Zd with d | e. It follows that T centralizes the full preimage C.D, which is a contradiction. ∼ Hence, C = R and Y = T × C = T × Zc.

Lemma 6.4.6. Let Γ be the graph constructed in Construction 6.4.2. If q ≥ 19, then AutΓ = Aq × Zc with c | e.

Proof. Let X ≤ AutΓ contain T . Then Γ is (X, 2)-path-transitive. Let R be the soluble radical of X. By Lemma 6.4.5, RT = R×T . It follows that X has a nonabelian minimal normal subgroup N say. Then N = Sl, where l ≥ 1 and S is a nonabelian simple group. Since T is simple, either T ≤ N, or T ∩ N = 1.

83 6.4. Proof of Theorem 6.1.4

Assume that T ∩ N = 1. Let Y = N:T . Then |N||Tα| = |Yα|. Suppose that N is transitive on V . Then (q − 2)! divides |N|. Let t be the largest prime which is no larger than q−2. Then t divides |N|, and it follows that tl divides |N|. l Since |N||Tα| = |Yα|, we have t divides |Yα|. However, by Theorem 6.1.2, Yα ≤ Sq ×Sq−1, so l ≤ 2. It follows that Y = N:T = N × T , which contradicts Lemma 6.4.4. Thus, N is not transitive on V . Furthermore, since T is simple and transitive on V we conclude that N has at least three orbits on V . Thus, N is semiregular on V and Γ is a cover of ΓN . By

Lemma 6.4.4, T does not centralise N. It follows that Aq = T ≤ Sl, and so l ≥ q. Since N is semiregular on V , and T is transitive, so |N| = |S|l divides |T | = q!/2, which is impossible. l Therefore, T ≤ N = S = S1 × · · · × Sl. Thus, T is a subgroup of some direct factor

Si, without loss of generality T ≤ S1 say. Suppose that l ≥ 2. Then N = TNα = S1Nα, l−1 ∼ ∼ and S = N/S1 = S1Nα/S1 = Nα/(S1)α. By Theorem 6.1.2, Xα has at most two insoluble composition factors, and hence l ≤ 3. Since |Γ (α)| = q, it follows that S ≤ Sq, 2 3 and so S = T = Aq and N = Aq or Aq. Now the order (q − 2)! = |V | = |N : Nα|, which 2 equals |Tα||Aq| or |Tα||Aq| , which is not possible. Thus l = 1, and N = S ≥ T is a simple group. Obviously, N is the only minimal normal subgroup of X which contains T .

Suppose that S > T . Then S = TSα = T1Sα, where T1 = NS(T ). Let U be a subgroup of S which contains T1 as a maximal subgroup. Then U acts transitively on V . Let B be a maximal U-invariant partition of V , and let B ∈ B. Then U B is primitive, and T1 is transitive on B. Thus, UB is a maximal subgroup of U, and we have a maximal factorization U = T1UB. If M is a minimal normal subgroup of U which is not a subgroup of T1, then T ∩ M = 1. Thus, T centralises M, and so M < T1, which is a contradiction. It follows that U has a unique minimal normal subgroup which contains

T1. Thus, U is an almost simple group, and U = T1UB is known by [79]. Inspecting the maximal factorisations of almost simple groups given in [79], and noticing that Aq = T ¢ T1 and q ≡ 3 (mod 4), we obtain that either

(a) U = An or Sn, where n = q + k with 1 ≤ k ≤ 5, and UB is a k-homogeneous group of degree n, or

1+2 (b) S = U = PSU(3, 5), T1 = T = A7, and UB = 5+ :8.

For the former, q + 1 divides |An|/|Aq|, and (q + k) ... (q + 1)q(q − 1)/2 divides |UB|.

Since U is transitive on V and Aq−2 < U, we have another factorisation U = Aq−2Uα.

By [79, Theorem D], Uα is a (k + 2)-homogeneous permutation group of degree n; in

84 6.4. Proof of Theorem 6.1.4

particular, Uα is 3-homogeneous. However, by Theorem 6.1.2, we have Uα ≤ Sq × Sq−1, which is not possible. For the latter, the order |V | = 5! = 60, but U has no subgroup of index 60, which is a contradiction.

Therefore, N = S = T and T is normal in AutΓ . Since q ≥ 19, Aut(T ) = Aq or Sq. It

follows from Lemma 6.4.4 that X = T ×C or (T ×C).Z2. Suppose that X = (T ×C).Z2. ∼ ∼ ∼ Then Xα/Tα = X/T = C.Z2, and X := X/C = Sq and ΓC is (X, 2)-path-transitive. Let ∼ α be the image of α in ΓC . Then Xα = Xα = Tα.C.Z2. It follows that C.Z2 = C × Z2, e ∼ e and Xα = Zp:(Tαβ × Z2).C = Zp:Zpe−1:Zc. In particular, Xαβ = Zpe−1:Zc. Then Xαβ = e 2 (Tαβ ×Z2).C = hsi:Zc ≤ ΓL(1, p ), where s = t. It follows that s = (1, 2, . . . q−2, q−1), a cycle of length pe − 1. However, the involution g deﬁned in Construction 6.4.2 does

not normalize hsi, which is not possible. So X = T × Zc, with c | e.

Proof of Theorem 6.1.4: Let q = pe ≡ 3 (mod 4) with p prime, and let Γ be the graph constructed in Construction 6.4.2. Assume that q ≥ 19. By Lemma 6.4.3, Γ is connected, and with g be chosen as in Construction 6.4.2, by Lemma 6.4.6, we have AutΓ = Aq × Zc, where c divides e. Thus, Γ is 2-path-transitive but not 2-arc- transitive. Then by Theorem 6.1.3, the line graph L(Γ ) is half-transitive, which is of valency 2(q − 1).

85 Chapter 7

Finite vertex-primitive and vertex-biprimitive 2-path-transitive graphs.

Basaed on the description of the local structures of 2-path-transitive graphs given in the last chapter, as another important application of our main result Theorem 4.1.1, in this chapter we deal with vertex-primitive and vertex-biprimitive 2-path-transitive graphs. For the primitive type of almost simple, we present a complete classification of such graphs. The classification contains the information of the full automorphism groups of the graphs. Furthermore, the numbers of the non-isomorphic graphs are determined. Combining the classification result with Lemma 6.3.1, we are able to construct some new half-transitive graphs in this chapter. This chapter is organised as follows. The main theorems are given in Section 1. A reduction for their proofs is given in section 2. Examples of the graphs appearing in the first two theorems are given in Section 3. Their proofs are given in Section 4.

7.1 The main results

As before we assume that Γ = (V,E) is a graph with vertex set V and edge set E, and G ≤ AutΓ . The vertex-primitive 2-path-transitive graphs are classiﬁed as follows.

Theorem 7.1.1. Let Γ be a G-vertex-primitive, (G, 2)-path-transitive graph of valency k. Assume that Γ is not (G, 2)-arc-transitive. Then k = pe ≡ 3(mod 4) with p prime, and one of the following holds:

d (i) G = Z2:Gα, where Gα is an irreducible subgroup of GLd(2), and is a 2-homogeneous permutation group of degree pe.

e (ii) G = PSL2(p ), and Γ = Kpe+1.

86 7.1. The main results

(iii) G = Ap, Gα = Zp:Z(p−1)/2, and k = p with p 6= 7, 11, 23; furthermore, either AutΓ = Ap, or AutΓ = Sp and Γ is 2-arc-regular.

(iv) AutΓ = G is a sporadic simple group, and (G, Gα) lies in the following table:

G Th B M

Gα Z31:Z15 Z47:Z23 Z59:Z29, Z71:Z35

The vertex-biprimitive 2-path-transitive graphs are given in the next theorem. Theorem 7.1.2. Let Γ be a connected (G, 2)-path-transitive graph of valency k which is not (G, 2)-arc-transitive. Assume further that Γ is G-vertex-biprimitive. Then Γ is bipartite, k = pe ≡ 3(mod 4) with p prime, and one of the following holds: (1) Γ is the standard double cover of a G+-vertex-primitive, (G+, 2)-path-transitive + + graph, where V Γ = ∆1 ∪ ∆2, G = G∆i , and G = G × Z2.

d (2) G = Zr:Gα:Z2, where r prime, d ≥ 1, and Gα is a 2-homogeneous permutation e group of degree p and is an irreducible subgroup of GLd(r).

e e e e (3) G = (Zp × Zp):Gαβ:Z2, where Gαβ ≤ ΓL1(p ) × ΓL1(p ), and Γ = Kpe,pe .

(4) AutΓ = G = Sp, Gα = Zp:Z(p−1)/2, where p ≡ 3(mod 4) is a prime and p 6= 7, 11, 23.

(5) G = PGL3(4).hτi with o(τ) = 2, Gα = (Z7:Z3) × Z3, k = 7, and AutΓ = Aut(PSL3(4)) = G.2;

(6) AutΓ = G = PΓU3(5), Gα = (Z7:Z3) × Z3, and k = 7. As mentioned before, one of our motivations for studying 2-path-transitive graphs is to construct new half-transitive graphs. The classiﬁcation given in Theorems 7.1.1 and 7.1.2 indeed leads to new constructions of half-transitive graphs. Theorem 7.1.3. For each group G and a subgroup H < G in TABLE C, there exists a half-transitive graph Σ of valency m such that AutΣ = G and H is the stabilizer in G of some vertex of Σ.

TABLE C

G H m Conditions 2 PΓU3(5) Z3:Z2 12 Th Z15:Z2 60 B Z15:Z2 60 Z23:Z2 92 M Z29:Z2 116 Z35:Z2 140 Ap Dp−1 2(p − 1) p ≡ 3 (mod 4) is a prime, and p 6= 7, 11 or 23. Sp Dp−1 2(p − 1) p is as above.

87 7.2. A reduction

7.2 A reduction

For a prime-power pe ≡ 3 (mod 4), it is easily shown that pe − 1 has primitive prime divisors. We thus have the following statement.

Corollary 7.2.1. If pe is the valency of a 2-path-transitive, 2-arc-intransitive graph, then pe − 1 has a primitive prime divisor which is at least 5.

Let Γ = (V,E) be a connected (G, 2)-path-transitive graph which is not (G, 2)-arc- transitive. Assume further that G is primitive or biprimitive on V . Let

+ G = hGα | α ∈ V i.

Then either G is primitive on V and G+ = G, or G is biprimitive on V , and G+ is of index 2 in G and has exactly two orbits on V which form the bipartition of V . In + + particular, in either case, Gα = Gα is maximal in G . The next lemma is a reduction for proving Theorems 7.1.1 and 7.1.2.

Lemma 7.2.2. Let Γ = (V,E) be a connected (G, 2)-path-transitive graph which is not (G, 2)-arc-transitive. Assume that G acts primitively or biprimitively on V . Then either e + Γ = Kpe,pe with p ≡ 3 (mod 4), or G is a primitive permutation group of aﬃne type or almost simple type.

Proof. Let Ω be an orbit of G+ on V , and let α ∈ Ω. By Theorem 6.1.1, the valency |Γ (α)| is pe for some prime p such that pe ≡ 3 (mod 4). Suppose that G+ acts unfaithfully on Ω. Then Γ is bipartite with parts Ω and Ω g, + + g + g where g ∈ G \ G . Since G is faithful on V = Ω ∪ Ω , the kernel G(Ω) acts on Ω + + + g + non-trivially. Since G(Ω) ¡ G and G is primitive on Ω , we conclude that G(Ω) is g transitive on Ω . It follows that Γ is a complete bipartite graph K|Ω|,|Ω g|, of valency |Ω|, and hence |Ω| = pe. Assume instead that G+ is faithful on Ω. Since G is primitive or biprimitive on + + V , we have G ≤ Sym(Ω) is primitive. By Theorem 6.1.1, the point stabilizer Gα is soluble, and hence in the language of the O’Nan-Scott Theorem (see [24]), the action of G+ on Ω is of aﬃne, almost simple, or product action type. + + Suppose that G is of product action type. Then G ≤ (T˜1 × T˜2 × · · · × T˜k):Sk k ≤ Sym(∆) o Sk, and Ω = ∆ , where k ≥ 2, T˜i ≤ Sym(∆) is almost simple with socle Ti, + ∼ k ∼ and T˜i is primitive on ∆. The socle, N = soc(G ) = T1 × · · · × Tk = T , where Ti = T , + + + + is a minimal normal subgroup of G . Since G = NGα , we know that Gα induces a + transitive action on {T1,T2,...,Tk} by conjugation. Since Gα is of odd order, it follows that k is odd and so k ≥ 3.

88 7.2. A reduction

∼ e By Theorem 6.1.1, the subgroup Op(Gα) = Zp is regular on Γ (α) and is a minimal

normal subgroup of Gα. Thus, either Op(Gα) ≤ Nα, or Op(Gα)∩Nα = 1. Further, since e Nα is transitive on Γ (α), the order |Nα| is divisible by p . If Op(Gα) ∩ Nα = 1, then e 2 |Gα| is divisible by (p ) , which is not possible by Theorem 6.1.1. Thus, Op(Gα) ≤ Nα,

and Nα = Op(Gα):Nαβ ¢ Gα. e Q Note that Zp:Nαβ = Op(Gα):Nαβ = Nα = Tiδi , where δi ∈ ∆. Suppose that m Nαβ = 1. Then Tiδi = Zp (for some m dividing e) is a maximal subgroup of the almost

simple group T˜i, which is impossible. Hence Nαβ 6= 1. Since Op(Gα) ∩ Nαβ = 1, there

exists a prime t 6= p which divides |Nαβ|. This prime t divides |Tiδi |, and it follows k that Nα contains a subgroup which is isomorphic to Zt . A Sylow t-subgroup of Nα is

isomorphic to a Sylow t-subgroup of Nα/Op(Gα), which is isomorphic to a subgroup of [1] [1] [1] [1] e Gαβ = (Gα × Gβ ).O. Since both Gα and Gαβ/Gα are subgroups of ΓL1(p ) which are metacyclic, it follows that k ≤ 4, and so k = 3. + Let L be the kernel of G acting on {T1,T2,T3} by conjugation. Then T1 ×T2 ×T3 = ∼ N ¡L ≤ T˜1 ×T˜2 ×T˜3, where Ti ≤ T˜i ≤ Aut(Ti) such that T˜i = L/CL(Ti). Since T1,T2,T3 + ∼ ∼ are conjugate in G , we have T˜1 = T˜2 = T˜3. Each element g ∈ L can be written as

g = (t1, t2, t3), for some ti ∈ T˜i.

Let πi be the projection from L to the i-th coordinate, namely, πi(g) = ti. Then ∼ πi(L) = T˜i.

Now X: = (T˜1 × T˜2 × T˜3) ≤ Sym(∆) o S3, and hence for a point α = (δ1, δ2, δ3) ∈ Ω, ˜ ˜ ˜ ˜ ˜ the stabilizer Xα is T1δ1 × T2δ2 × T3δ3 . Since πi(L) = Ti, we have πi(Lα) = Tiδi . By e e Lemma 7.2.1, p − 1 has a primitive prime divisor r ≥ 5. Since (p − 1)/2 divides |Gα|, + ∼ + so does r. Further, as Gα /Lα = G /L ≤ S3, we conclude that r | |Lα|, and so r divides ˜ |Tiδi |. e ∼ By Theorem 6.1.1, Zp = Op(Gα) ¡ Nα ¡ Gα, and hence

l ∼ ∼ ˜ Zp = πi(Op(Gα)) ¡ πi(Nα) = Tiδi ¡ Tiδi ,

˜ l where l = e/3. We may write Tiδi = Zp:Ki. Then r divides |Ki|.

Let Ci be a Sylow r-subgroup of Ki, and consider the subgroup Mi: = πi(Op(Gα)):Ci ˜ ∼ l of Ti. If Ci acts non-trivially on πi(Op(Gα)) = Zp, then Ci < GLl(p), which contradicts l the fact that r - p − 1. Thus Mi = πi(Op(Gα)) × Ci. Notice that a Sylow r-subgroup of Gα is contained in a Sylow r-subgroup of T˜1 × T˜2 × T˜3, from which it follows that all [1] the r-elements of Gα are in the centralizer of Op(Gα), which is Gα by Theorem 6.1.1. Γ(α) Since r does not divide |Gα |, we again obtain a contradiction.

89 7.3. Examples

Hence G+ is not of product action type, that is, the primitive type of G+ is aﬃne or almost simple.

7.3 Examples

In this section, we construct and study examples of 2-path-transitive graphs appearing in Theorems 7.1.1 and 7.1.2. Let G be a ﬁnite group and let H be a core-free subgroup of G. We restate the deﬁnition of coset graphs as follows. Denote by [G : H] the set of right cosets of H in G, namely, [G : H] = {Hx | x ∈ G}. For an element g ∈ G with g2 ∈ H, the coset graph of G with respect to H and g is the graph with vertex set [G : H] such that Hx and Hy are adjacent if and only if yx−1 ∈ HgH. This coset graph is G-arc-transitive and is denoted by Cos(G, H, HgH). Assume that Γ = Cos(G, H, HgH) is a (G, 2)-path-transitive graph which is not (G, 2)-arc-transitive. Denote by α, β the vertices corresponding to H and Hg, respec- g 2 2 tively. Then (α, β) = (β, α), and g ∈ Gα. Since |Gα| is odd, we have g = 1, that is, g is an involution. Also Γ is connected if and only if hH, gi = G. To construct 2-path-transitive graphs which are not 2-arc-transitive, we need the following result, the proof of which is straightforward.

Lemma 7.3.1. Let G be a ﬁnite group with a core-free subgroup H and an involution g. Assume further that hH, gi = G, and the coset action of H on [H : H ∩ Hg] is 2-homogeneous but not 2-transitive. Then the coset graph Cos(G, H, HgH) is (G, 2)- path-transitive but not (G, 2)-arc-transitive.

e Example 7.3.2. Let Γ = Kq+1, where q = p ≡ 3(mod 4), with p prime. Then

Aut(Γ ) = Sq+1 contains a subgroup G = PSL2(q). For a vertex α of Γ , we have e Gα = Zp:Z(pe−1)/2, which is a maximal subgroup of G. Thus by Lemma 7.3.1, the graph Γ is G-vertex-primitive, (G, 2)-path-transitive but not (G, 2)-arc-transitive.

e Example 7.3.3. Let Γ = Kpe,pe , where p ≡ 3(mod 4), be the complete bipartite graph e e e of order 2p . Then Aut(Γ ) = Spe o Z2. Let G = ((Zp × Zp):H):Z2 < Aut(Γ ), where e Z(pe−1)/2 ≤ H ≤ ΓL1(p ), and |H| is odd. Then the graph Γ is G-vertex-biprimitive, (G, 2)-path-transitive but not (G, 2)-arc-transitive. The primitive type of the index two subgroup G+ of G is of aﬃne.

Next, we study a family of 2-path-transitive graphs associated with alternating groups.

Example 7.3.4. Let G = Sp, and T = Ap, both acting naturally on the set {1, 2, . . . , p}, where p is prime, such that p ≡ 3(mod 4), with p 6= 7, 11, 23. Let H be a subgroup of T isomorphic to K:L, where K = Zp and L = Z(p−1)/2. Then H is a maximal subgroup

90 7.3. Examples

of T . Since subgroups which are isomorphic to H are conjugate in T , it follows that L 2 is conjugate to ht i, where t is a (p − 1)-cycle. Let g ∈ NT (L)\L and f ∈ NG(L)\Ap be involutions. Deﬁne

Γ = Cos(T, H, HgH) and Σ = Cos(G, H, HfH).

For example, we may choose t = (1, 2, . . . , p − 1). Then

t2 = (1, 3, . . . , p − 2)(2, 4, . . . , p − 1),

and g and f can be deﬁned as follows:

p−1 p+3 p+1 p+5 g = (3, p − 2)(5, p − 4) ... ( 2 , 2 )(4, p − 1)(6, p − 3) ... ( 2 , 2 ) ∈ Aq, f = (1, 2)(3, 4) ... (p − 2, p − 1) ∈ Sq \ Aq.

By the Construction 6.4.2, the graph Γ is connected, (T, 2)-path-transitive but not (T, 2)-arc-transitive, and AutΓ = T or G. Further, since H is a maximal subgroup of T , Γ is T -vertex-primitive. For the graph Σ, we have the following conclusion.

Lemma 7.3.5. Let Σ be the graph constructed in Example 7.3.4. Then Σ is connected, vertex-biprimitive, and 2-path-regular; furthermore, AutΣ = Sp.

∼ Proof. By the deﬁnition, f ∈ G \ T normalizes L. Since H < T and G/T = Z2, T has + exactly two orbits on [G : H], say ∆1 and ∆2, such that G : = G∆i = T . Moreover, the + setwise stabilizer G is primitive on ∆i, and Σ is a bipartite graph with parts ∆1 and f ∆2. Since f does not normalize H, it follows that Σ is connected. Since H ∩ H = L, and the action of H on [H : H ∩Hf ] is 2-homogeneous but not 2-transitive, we conclude that Σ is a (G, 2)-path-transitive graph of valency p which is not (G, 2)-arc-transitive. Further, Γ is G-vertex-biprimitive. Suppose that G+ is not normal in AutΣ. Then there exists a group X such that

T < X ≤ AutΣ, and NAutΣ(T ) is a maximal subgroup of X. Since T is primitive on ∆i, so is X. Since |∆i| = |T : H| = (p − 2)!, by the O’Nan-Scott Theorem, we conclude that either X is almost simple or X£Ap−2×Ap−2. Moreover, because the order |X| is divisible by p, it follows that X is almost simple. Thus we have X = NAutΓ (T )Xα is a maximal factorisation of the almost simple group X. Therefore the triple (X, NAutΓ (T ),Xα) should lie in the classiﬁcation of [79]. However, an inspection shows that there is no such triple, which is a contradiction. Thus, G+ = T is normal in AutΣ. Since T is primitive on ∆i, the centralizer of T in AutΣ is trivial, so AutΣ ≤ Aut(T ) = Sp. Since

AutΣ ≥ G = Sp, we have AutΣ = Sp.

91 7.3. Examples

Let Γ = Cos(G, H, HgH), and let Aut(G, H) = {σ ∈ Aut(G) | Hσ = H}. The following conclusion is well known and the proof is easy.

Lemma 7.3.6. Suppose that σ ∈ Aut(G, H). Then Γ = Cos(G, H, HgH) is isomorphic to Σ = Cos(G, H, HgσH). Moreover, σ induces an automorphism of Γ if and only if HgH = HgσH.

Let Γ be the graph and t be a (p − 1)-cycle deﬁned in Example7 .3.4. Let M = hti. We point out that the choice of g for deﬁning Γ is not unique. Indeed we can even choose g such that g ∈ NT (L)\NT (M). This is because NT (L) is an index two subgroup of NG(L), and further, as NT (M) ≤ NT (L), and (p − 1)/2 is odd, we ∼ have NG(M) = M:Aut(M) = Zp−1:Aut(Zp−1) = Zp−1:Aut(Z(p−1)/2). Since NT (L) ≥

Zp−1:(Aut(Z(p−1)/2)×Aut(Z(p−1)/2)), we conclude that NT (M) < NT (L), and a Sylow-2 subgroup of NT (M) is properly contained in a Sylow-2 subgroup of NT (L). Therefore, there exists a 2-element g such that g ∈ NT (L)\NT (M). Actually the full automorphism group of Γ depends on the choice of g.

The case AutΓ = Sp does occur.

Lemma 7.3.7. If g ∈ NT (M)\M, then AutΓ = Sp and Γ is 2-arc-regular.

Proof. Let H1 = K:M and let V1 = [G : H1]. Then H is an index 2 subgroup of H1.

Now the graph Γ1 = Cos(G, H1,H1gH1) is 2-arc regular, with full automorphism group

Sp (see [32, Lemma 4.2]). Further, since T is transitive on V1, for any H1x ∈ V1, we may choose x ∈ T . Then it is straightforward to show that the map ψ : V1 7→ V deﬁned by

ψ(H1x) = Hx is a graph isomorphism between Γ1 and Γ . It follows that AutΓ = Sp, and Γ is 2-arc-regular.

The case AutΓ = Ap occurs too.

Lemma 7.3.8. If g ∈ NT (L)\NT (M), then AutΓ = Ap and Γ is 2-path-regular.

Proof. As indicated in Example 7.3.4, AutΓ = Ap or Sp. Suppose that AutΓ = Sp. σ Write M = L × hσi, where o(σ) = 2. Then hAp, σi = Sp and H = H. Thus by Lemma σ σ σ 7.3.6, we have HgH = Hg H. Since g∈ / NT (M), we obtain g 6= g. Since g ∈ HgH σ σ and g ∈ NT (L), there exist a, b ∈ K and u ∈ L such that g = agbu. As o(g ) = 2, we have (agbu)2 = 1, so (bua)g = (bua)−1. Since bua ∈ H, we obtain bua ∈ H ∩ Hg = L. −1 u u σ a−1 As u ∈ NT (K), we have u bua = b a ∈ K ∩ L, that is, b a = 1. Thus g = (gu) . Since gσ normalizes L, we have (La)gu = La. Notice that La ∈ H, so La ≤ H ∩Hgu = L, and therefore La = L. If a 6= 1, then the order of a is p, which means that K normalizes L, which is a contradiction. Thus a = b = 1 and gσ = gu. Since both g and σ are

92 7.3. Examples involutions, we obtain σg = uσ, that is, g normalizes M, which is again a contradiction.

Thus AutΓ = Ap, as claimed.

The graphs in the next example arise from three sporadic simple groups: the Thomp- son group Th, the Baby Monster B, and the Monster M.

Example 7.3.9. Let G = Th, B or M. Then by the Atlas [19], the group G contains ∼ a maximal subgroup H such that H = Zp:Z(p−1)/2, where p = 31 for G = Th, p = 47 for G = B, and p = 59 or 71 for G = M. Let L be a subgroup of H which is isomorphic to Z(p−1)/2. Then in each case, NG(L) is of even order. This is true because the Atlas [19] shows that for G = Th, |CG(L)| = 30; for G = B, |CG(L)| = 46; for G = M with p = 59, NG(L) = (Z29:Z14 × 3).2; and for G = M with p = 71, |CG(L)| = 70 or 2100. g Let g be an involution in NG(L), and let Γ = Cos(G, H, HgH). Then H ∩ H = L, and hence Γ has valency |H : L| = p. Since H is a maximal subgroup of G, we have hH, gi = G, and thus Γ is connected. Further, since the action of H on [H : L] is 2-homogeneous and is not 2-transitive, we conclude that Γ is G-vertex-primitive, (G, 2)- path-transitive but (G, 2)-arc-intransitive.

Moreover, for the graphs that we just constructed in Example 7.3.9, the following conclusion is true.

Lemma 7.3.10. If Γ is a graph in Example 7.3.9, then AutΓ = G and Γ is 2-path- regular.

Proof. Suppose that G is not normal in AutΓ . Then there exists a group X such that

G < X ≤ AutΓ and NAutΓ (G) is a maximal subgroup of X. Since G is primitive on V Γ , so is X. Also because Γ is (G, 2)-path-transitive, we have either Γ is (X, 2)-arc- transitive, or Γ is (X, 2)-path-transitive but not (X, 2)-arc-transitive. Thus the primitive type of X is affine, almost simple, product action or twisted wreath product. Notice that |V Γ | = |G:H| is not a power of an integer, and |G:H| is exactly divisible by 19, from which we conclude that X is almost simple. Thus, X = NAutΓ (G)Xα is a maximal factorisation, and hence the triple (X, NAutΓ (G),Xα) lies in the classification given in [79]. However, an inspection of the classification shows that there is no such a triple, which is a contradiction. Therefore, G is normal in AutΓ . Since G is primitive on V , the centralizer of G in AutΓ is trivial. As Out(G) = 1, we conclude that AutΓ = G and Γ is 2-path-regular.

The following two examples arise from classical groups of Lie type.

Example 7.3.11. Let T = PGL3(4). Then by the Atlas [19], T contains a maximal subgroup H which is isomorphic to (Z7:Z3) × Z3. Let G = PGL3(4).hτi, where τ is a ﬁeld automorphism or the graph automorphism of PSL3(4) of order 2. Write H = K :

93 7.3. Examples

L = (hxi:hyi) × hzi, where K = hxi, L = hyi × hzi, o(x) = 7, o(y) = o(z) = 3, and z is a diagonal automorphism of PSL3(4). Since subgroups of G which are isomorphic to

H are conjugate, we may assume that x, y ∈ PGL3(2). Then zx = xz and zy = yz.

By the Atlas [19], NG(K) = (7:3 × 3).2. Since xy 6= yx, we have |CG(K)| = 21 or 42. If |CG(K)| = 42, then NG(K) contains only three involutions. Assume that |CG(K)| = 21. Then NG(K)/CG(K) = Z6. Thus there are at most 21 involutions in NG(K). Suppose that a ∈ NG(K) is an involution such that a ∈ NG(L). We claim xk that none of the involutions a , where 1 ≤ k ≤ 6, is contained in NG(L). Assume that xk xk a ∈ NG(L). Then aa ∈ NG(L). By deﬁnition, hx, ai is a dihedral group of order xk 14, and so aa has order 7. Since |NT (L)| = 27, we have |NG(L)| is 27 or 54. Since |NG(L)| is even, we have |NG(L)| = 54. But 7 - |NG(L)|, a contradiction. It follows that among the involutions of NG(K), at most three of them are contained in NG(L). Thus there exists an involution g ∈ NG(L)\NG(K), and the graph Γ = Cos(G, H, HgH) is G-vertex-biprimitive, (G, 2)-path-transitive but not (G, 2)-arc-transitive, and of valency 7.

Example 7.3.12. Let T = PGU3(5). Then by the Atlas [19], T contains a maximal subgroup H which is isomorphic to (Z7:Z3) × Z3. Let G = PΓU3(5). As in Example 7.3.11, write H = K:L. Then by GAP (see http://www.gap-system.org/), NG(L) contains nine involutions, three of which are contained in NG(K). Thus for any 2- element g ∈ NG(L)\NG(K), the graph Γ = Cos(G, H, HgH) is G-vertex-biprimitive, (G, 2)-path-transitive but not (G, 2)-arc-transitive, of valency 7.

For the graphs in Examples 7.3.11 and 7.3.12, the full automorphism groups of the graphs are determined in the next two lemmas.

Lemma 7.3.13. If Γ is the graph in Example 7.3.11, then AutΓ = PSL3(4).(2 × S3) and Γ is 2-arc-transitive.

Proof. By the definition, the group G = PGL3(4).hτi, where τ is a field automorphism or the graph automorphism of PSL3(4) of order 2. Assume first that τ is the graph automorphism of PSL3(4) (that is, τ is the transpose inverse map). Now

Γ = Cos(G, Gα,GαgGα), with Gα = 7:3 × 3. Let T = PGL3(4) and denote Gα = K : L = (hxi:hyi) × hzi, where K = hxi, L = hyi × hzi, o(x) = 7, o(y) = o(z) = 3, and z is a diagonal automorphism of PSL3(4). Then the 2-element g lies in NG(L)\NG(K). σ Let σ be a ﬁeld automorphism of PSL3(4). We claim that g ∈ HgH.

As observed in Example 7.3.11, the normaliser NT (L) is a Sylow 3-subgroup of G, with |NT (L)| = 27 and |NG(L)| = 54. Thus we may write NT (L) = (hyi×hzi):hti, where k k k o(t) = 3. Then g can be written as g = g1τ, where g1 = y 1 z 2 t 3 ∈ T , with ki = ±1. As both g and τ normalize L, so does g1. Since τ normalizes H, we have HgH = Hg1τH = σ σ σ Hg1Hτ. Because στ = τσ, we have g = g1 τ, and hence g ∈ HgH = Hg1Hτ if and σ only if g1 ∈ Hg1H. By the Atlas [19], σ ﬁxes the Sylow 3-subgroup NT (L), and thus

94 7.3. Examples

σ σ l1 l2 l3 l1 l2 l3−k3 k3 g1 ∈ NT (L). Assume that g1 = y z t = y z t t . Since t normalizes hyi × hzi, σ it follows that g1 ∈ Hg1H. Thus σ induces an automorphism of Γ . If we interchange the roles of τ and σ by assuming that τ is a ﬁeld automorphism of PSL3(4), and σ is the graph automorphism, then analogously we have σ ∈ AutΓ . It follows that in both ∼ cases, AutΓ ≥ PGL3(4).(hτi × hσi) = PSL3(4).(2 × S3). We will show that the equality holds. Let A = PSL3(4).(2 × S3).

Suppose, to the contrary, that X0 := AutΓ > A. Then there exists a subgroup X of X0 such that A < X ≤ X0 and A is maximal in X. We have two possibilities: X is primitive or biprimitive on V . Assume that X is primitive on V . Then Γ is (X, 2)-path-transitive (possibly (X, 2)- arc-transitive). Thus X is of type affine, almost simple, product action, or twisted wreath product. Assume first that X is of type affine. Then for any α ∈ V , Xa is faithful on Γ (α). On the other hand, Aα ≤ Xα, both Aα and Xα act unfaithfully on Γ (α), and we obtain a contradiction. Since |V | = 1920, we know that X is neither product action type nor twisted wreath product type. Thus X is almost simple, and X = AXα is a maximal factorisation. Hence the triple (X, A, Xα) lies in the classification given in [79].

Notice that |V | = |X:Xα| = 1920, so that an inspection of the classiﬁcation shows that there is no such a triple, which is a contradiction. Assume that X is biprimitive on V . Then the two invariant blocks of X are the same as that of G, which we suppose to be ∆ and ∆0. Then X induces a primitive action on both ∆ and ∆0. Since |V | = 1920, X is not of type aﬃne, product action, or twisted wreath product. Thus X∆ is almost simple, and once again we obtain a maximal factorisation X∆ = (A ∩ X∆)(X∆)α, where A ∩ X∆ = PGL3(4).2. An inspection of [79] shows that there is no such a factorisation, a contradiction.

From the above discussion, we come to the conclusion that AutΓ = PSL3(4).(2 × S3) and Γ is 2-arc-transitive, as claimed.

Lemma 7.3.14. If Γ is the graph in Example 7.3.12, then AutΓ = G and Γ is 2-path- transitive and 2-arc-intransitive.

Proof. Since AutΓ ≥ G, we only need to show that equality holds. Suppose, on the contrary, that X0 := AutΓ > G. Then there exists a subgroup X of X0 such that

G < X ≤ X0, and G is maximal in X. Now we have |V | = 12000, and the argument in Lemma 7.3.13 also applies to the current case. Thus for the graph Γ in Example 7.3.12, analogously we have AutΓ = G.

95 7.4. Proof of the main theorems

7.4 Proof of the main theorems

This section is devoted to complete the proofs of our main theorems in this chapter. The proofs of the Theorem 7.1.1 and 7.1.2 depend on the information about almost simple primitive permutation groups of which the point stabiliser has odd order, which can be easily distilled from TABLE I– TABLE VII, given as follows.

Proposition 7.4.1. Let G be an almost simple group with T = soc(G). Assume that M is a maximal subgroup of G of odd order. Then one of the following holds:

(i) G = Ap, and M = Zp:Z(p−1)/2, where p ≡ 3 (mod 4) is a prime, and p 6= 7, 11 or 23;

e e e (ii) T = PSL2(p ), and T ∩M = Zp:Z(pe−1)/2, where p is a prime, and p ≡ 3 (mod 4);

(iii) T = PSLr(q), and T ∩ M = Z(qr−1)/(q−1)(r,q−1):Zr, where r is an odd prime;

(iv) G = PSL3(4).3, and M = (Z7:Z3) × Z3;

(v) T = PSUr(q), and T ∩ M = Z(qr+1)/(q+1)(r,q+1):Zr, where r is an odd prime, and (r, q) 6= (3, 3), (5, 2) or (3, 5);

(vi) G = PSU3(5).3, and M = (Z7:Z3) × Z3.

(vii) G and M lie in the following table:

G M23 Th B M M Z23:Z11 Z31:Z15 Z47:Z23 Z59:Z29, Z71:Z35

We remark that the above classiﬁcation had also been obtained by Liebeck and Saxl in [81, Theorem 2]. As shown in the examples of the last section, almost all of the pairs (G, M) in Theorem 7.4.1 do give rise to (G, 2)-path-transitive graphs which are G-vertex-primitive or G-vertex-biprimitive.

7.4.1 Proof of Theorem 7.1.1

Let Γ be a G-vertex-primitive and (G, 2)-path-transitive and (G, 2)-arc-intransitive e graph, of valency k. Then Gα is primitive on Γ (α) and by Theorem 6.1.1, k = p ≡ 3(mod 4) with p prime. By Lemma 7.2.2, G is of type aﬃne or almost simple. Assume n that G is of type aﬃne, with N: = soc(G) = Zp . Then we may identify V Γ with N such that α is the zero vector. Let β ∈ Γ (α). Then the pair {β, −β} is a block of imprimitivity in Γ (α) of Gα. Since Gα is 2-homogeneous on Gα, we have β = −β, so p = 2, as in (1) of Theorem 7.1.1. Assume that G is almost simple, and T ≤ G ≤ Aut(T ), with T

96 7.4. Proof of the main theorems

a non-abelian simple group. Then the pair (G, Gα) appears in Theorem 7.4.1. Thus we need to analyse the pairs of Theorem 7.4.1 in turn.

(1) Assume that (G, Gα) = (Ap, Zp:Z(p−1)/2), (Th, Z31:Z15), (B, Z47:Z23), (M, Z59:Z29), e e or (M, Z71:Z35), or (T,Tα) = (PSL2(p ), Zp:Z(pe−1)/2). Then by Examples 7.3.2, 7.3.4, and 7.3.9, each of these cases corresponds to a G-vertex-primitive, (G, 2)-path-transitive but not (G, 2)-arc-transitive graph, as in Theorem 7.1.1.

∼ (2) Assume that (G, Gα) = (M23, Z23:Z11). We write Gα as K:L, where K = Z23, ∼ L = Z11. Then Gα is the normalizer of a Sylow-23 subgroup of M23. By the Atlas ∼ [19], the cyclic subgroups of order 11 form two conjugacy classes in G. For L = Z11, assume that the order |NG(L)| is even. Then NG(L) is contained in a maximal subgroup isomorphic to M11 or M22. Suppose that NG(L) < M, where M is M11 or M22. Then it follows that NG(L) = NM (L), and by the Atlas [19], NM (L) = Z11:Z5, a contradiction. Thus there is no (G, 2)-path-transitive but not (G, 2)-arc-transitive graph occuring in this case.

r (3) Assume that (T,Tα) = (PSLr(q), Z(qr−1)/(q−1)(r,q−1):Zr). Denote m = (q − 1)/(q − 1)(r, q − 1). Then

Gα = (Zm:Zr).o where o = G/T . By Theorem 6.1.1, the unique minimal normal subgroup of Gα is an elementary abelian group, so m is a prime, and r = (m−1)/2. Notice that if q = pe with p ≥ 3, then r < (m−1)/2. Thus p = 2, (r, q) = (3, 4), and (G, Gα) = (PGL3(4), 7:3×3). If there exists a 2-path-transitive and 2-arc-intransitive graph Γ , let (α, β) be an arc of Γ . Then Γ = Cos(G, Gα,GαgGα), with the 2-element g interchanging α and β.

Assume that Gα = (hxi:hyi) × hzi, where o(x) = 7, o(y) = o(z) = 3, and z is a diagonal [1] [1] [1] [1] automorphism. Then Gα = hzi, Gβ = hyi, and g interchanges Gα with Gβ . Checking 2 the Atlas [19], L := hyi × hzi is contained in a subgroup isomorphic to Z3:2A4. Thus 2 y ∈ Z3, and z ∈ 2A4, but it is easy to see that no 2-element of 2A4 interchanges hyi with

hzi, a contradiction (actually by GAP, NG(L) = 27). Thus in this case there exists no (G, 2)-path-transitive but not (G, 2)-arc-transitive graph.

(4) Assume that (T,Tα) = (PSUr(q), Z(qr+1)/(q+1)(r,q+1):Zr). Then by a similar

argument to the one above, the only possibility is (G, Gα) = (PGU3(5), 7:3 × 3). De- note by (hxi:hyi) × hzi the subgroup 7:3 × 3, where o(x) = 7, o(y) = o(z) = 3, and let L = hyi×hzi. Using GAP (here we use the “pq-package”, see http://cage.ugent.be/ jde-

beule/pg.html), then calculation shows that |NG(L)| = 27. Thus there exists no 2- element g which satisﬁes the conditions of Lemma 7.3.1, so no such graph exists. This completes the proof of Theorem 7.1.1.

97 7.4. Proof of the main theorems

Let Γ be a ( directed or undirected) graph with vertex set V . Then the standard double cover of Γ is deﬁned to be the undirected bipartite graph Γ˜ with parts V0 and

V1, where Vi = {(v, i) | v ∈ V }, such that two vertices (x, i) and (y, j) are adjacent if and only if i 6= j, and x, y are adjacent in Γ .

Lemma 7.4.2. Let Γ = Cos(G, H, HgH) be a G-vertex-primitive, (G, 2)-path-transitive and (G, 2)-arc-intransitive graph. Let Σ = Cos(K,H1,H1(g, z)H1), where K = G × hzi,

H1 = H × {1}, and z is an involution. Then

(1) Σ is isomorphic to the standard double cover of Γ ;

(2) Σ is K-vertex-biprimitive, (K, 2)-path-transitive and (K, 2)-arc-intransitive.

Proof. (1) We have that V Γ = [G:H] = {Hx | x ∈ G}. Let Γ˜ denote the standard double cover of Γ . Then V Γ˜ = {(Hx, i) | x ∈ G, i = 0 or 1}. Deﬁne a map ψ : V Γ˜ 7→ V Σ as follows: i ψ((Hx, i)) = H1(x, z ).

This map is clearly one-to-one. Notice that z−i = zi, for x, y ∈ G and i, j ∈ {0, 1}, so we have

{(Hx, i), (Hy, j)} ∈ EΓ˜ ⇐⇒ yx−1 ∈ HgH, and i + j = 1 −1 i+j ⇐⇒ (yx , z ) ∈ H1(g, z)H1 j i −1 ⇐⇒ (y, z )(x, z ) ∈ H1(g, z)H1 i j ⇐⇒ {(H1(x, z ), (H1(y, z )} ∈ EΣ.

Therefore, Γ˜ =∼ Σ. Γ(α) (2) Let α = H ∈ V Γ , and let α1 = H1 ∈ V Σ. Then it is easy to show that Gα Σ(α1) is permutationally isomorphic to Kα1 . It follows that Σ is K-vertex-biprimitive, (K, 2)-path-transitive and (K, 2)-arc-intransitive.

7.4.2 Proof of Theorem 7.1.2

Let Γ be a G-vertex-biprimitive, (G, 2)-path-transitive and (G, 2)-arc-intransitive graph, 0 with two parts ∆ and ∆ . Then by Lemma 7.2.2, the primitive type of G∆ is aﬃne, or almost simple. If CG(G∆) is non-trivial, then since |G : G∆| = 2, we have G =

G∆ × Z2. Therefore Γ is the standard double cover of one of the graphs in Theorem 7.1.1. Combining this with Lemma 7.4.2, we come to the conclusion (1) of Theorem

7.1.2. Thus we assume that CG(G∆) = 1.

Assume ﬁrst that G∆ is aﬃne. We consider two cases: G∆ is faithful on ∆ and G∆ is unfaithful on ∆.

Suppose that G∆ is faithful on ∆. Since G∆ is not regular on ∆ and CG(G∆) = 1, ∼ d we have soc(G) = soc(G∆) = Zr with r prime. Therefore for α ∈ ∆, there exist a prime

98 7.4. Proof of the main theorems

e e number p and an odd number e such that p ≡ 3(mod 4), and Gα = (G∆)α < AΓL1(p ) is 2-homogeneous but not 2-transitive of degree pe, and is an irreducible subgroup of

GLd(r), as in (2) of Theorem 7.1.2.

Suppose that G∆ is unfaithful on ∆. Since Γ is G-vertex-biprimitive and the kernel 0 e 0 G(∆) is transitive on ∆ , it follows that Γ = Kpe,pe , where p = |∆| = |∆ |, with p prime, 0 ∆0 e odd. For α ∈ ∆, we have Γ (α) = ∆ , and Gα is 2-homogeneous but not 2-transitive, ∆0 e e e so Gα = Zp:Gαβ. By Theorem 6.1.1, we have Gαβ ≤ ΓL1(p ) × ΓL1(p ). It follows that e e G = (Zp × Zp):Gαβ:Z2, as in (3) of Theorem 7.1.2.

Assume next that G∆ is almost simple. Then the pair (G∆, (G∆)α) = (G∆,Gα) appears in Theorem 7.4.1. Thus we need to consider all the candidates in Theorem 7.4.1.

(a) Assume that soc(G∆) = Ap with p ≥ 5. Then (G, Gα) = (Sp, Zp:Z(p−1)/2), with p prime, p ≡ 3(mod 4) and p 6= 7, 11, 23. By Example 7.3.4 and Lemma 7.3.5, there exists

a graph Σ = Cos(G, Gα,GαfGα), which is G-vertex-biprimitive, (G, 2)-path-transitive

but not (G, 2)-arc-transitive, with valency k = p. Further, AutΣ = Sp, as indicated in Theorem 7.1.2.

(b) Assume that soc(G∆) is a sporadic almost simple group. Then by Theorem 7.4.1,

soc(G∆) = M23, Th, B or M. For each case, since CG(G∆) = 1, and Out(soc(G∆)) = 1, we obtain a contradiction. Thus there is no G-vertex-biprimitive, (G, 2)-path-transitive but (G, 2)-arc-intransitive graph arising in this case.

(c) Assume that soc(G∆) is a classical group of Lie type. Then by Theorem 7.4.1, we need to consider the following cases: e e either (soc(G∆), soc(G∆)α) = (PSL2(q), Zp:Z(pe−1)/2), where q = p ≡ 3(mod 4), or

(G, Gα) = (PGL3(4).hτi, 7:3 × 3) or (PGU3(5).hτi, 7:3 × 3), where τ is an involution. e Assume that (soc(G∆), soc(G∆)α) = (PSL2(q), Zp:Z(pe−1)/2). Then G = PSL2(q).2 e and Gα = soc(G∆)α = Zp:Z(pe−1)/2 = K:L, say. Let T := PSL2(q). Then by a result of ∼ e ∼ ∼ Dickson in 1901 (see [52]), NG(K) = Zp:Z(pe−1), NG(L) = D2(q−1), and NT (L) = D(q−1). q−1 Write NG(L) = hxi:hδi, where o(x) = q − 1, and o(δ) = 2. Let g := x 2 δ. Since δ ∈ T

and δ does not normalize K, we have g ∈ G\T is a 2-element, and g ∈ NG(L)\NG(K).

Thus the graph Γ = Cos(G, Gα,GαgGα) is G-vertex-biprimitive, (G, 2)-path-transitive but (G, 2)-arc-intransitive. Since the order of Γ is 2(q + 1) and the valency of Γ is q,

we conclude that Γ = Kq+1,q+1 − (q + 1)K2. Thus Γ is the standard double cover of the

complete graph Kq+1, and therefore (1) holds.

Assume that (G, Gα) = (PGL3(4).hτi, 7:3×3). Then by Example 7.3.11, there exists a graph Γ = Cos(G, Gα,GαgGα) which is G-vertex-biprimitive, (G, 2)-path-transitive but (G, 2)-arc-intransitive. Further, by Lemma 7.3.13, AutΓ = G.2, as shown in (5) of Theorem 7.1.2.

99 7.5. The numbers of the non-isomorphic graphs

Assume that (G, Gα) = (PGU3(5).hτi, 7:3×3). Then by Example 7.3.12, there exists a graph Γ = Cos(G, Gα,GαgGα) which is G-vertex-biprimitive, (G, 2)-path-transitive but (G, 2)-arc-intransitive, with AutΓ = G, as shown in (6) of Theorem 7.1.2. This completes the proof of Theorem 7.1.2.

Recall that the line graph L(Γ ) of a graph Γ = (V,E) is the graph with vertex set E such that two vertices in L(Γ ) are adjacent if and only if they are incident with some vertex α ∈ V . By Theorem 6.1.3, a graph is 2-path-transitive but 2-arc-intransitive if and only if its line graph is half-transitive.

7.4.3 Proof of Theorem 7.1.3

Let G be a group given in the ﬁrst column of TABLE C. Then, by Theorems 7.1.1 and 7.1.2, there exists a connected (G, 2)-path-transitive graph Γ such that AutΓ = G, the vertex-stabilizer Gα, and the valency k lie in the following table, where p ≡ 3 (mod 4) and p 6= 7, 11 or 23:

G PΓU3(5) Th B M Ap Sp Gα (Z7:Z3) × Z3 Z31:Z15 Z47:Z23 Z59:Z29, Z71:Z35 Zp:Z(p−1)/2 Zp:Z(p−1)/2 k 7 31 47 59, 71 p p

Let Σ be the line graph of Γ . Then the valency of Σ equals 2(k − 1), which is the value of m given in the third column of TABLE C. By Theorem 6.1.3, a graph is 2-path-transitive but not 2-arc-transitive if and only if its line graph is half-transitive. Since the graph Γ has more than 5 vertices, a result of Whitney (1932) (see [3]) implies that AutΣ =∼ AutΓ = G. It follows that the line graph Σ is half-transitive. An edge

{α, β} of Γ is a vertex of Σ, denoted by v. Thus, the vertex-stabilizer Gv for Σ is the stabilizer of the edge {α, β}. In particular, the vertex-stabilizer Gv is the subgroup H listed in the second column of TABLE C.

7.5 The numbers of the non-isomorphic graphs

Let Γ be a G-vertex-primitive or G-vertex-biprimitive, (G, 2)-path-transitive but not

(G, 2)-arc transitive graph, with G almost simple. Then the pair (G, Gα) is given in Theorem 7.1.1 or Theorem 7.1.2, as listed in the following:

e (Ap, Zp:Z(p−1)/2), (Sp, Zp:Z(p−1)/2), (PSL2(q), Zp:Z(pe−1)/2),

(Th, Z31:Z15), (B, Z47:Z23), (M, Z59:Z29), (M, Z71:Z35), (7.5.1)

100 7.5. The numbers of the non-isomorphic graphs

(PGL3(4).hτi, 7:3 × 3), (PΓU3(5), 7:3 × 3)

Our aim here is to determine the number of the non-isomorphic graphs corresponding e to each pair. For (G, Gα) = (PSL2(q), Zp:Z(pe−1)/2), since the graph is Kpe+1, it is unique up to isomorphic. So we only need to consider the other pairs. Here the strategy is that of [32], where the non-isomorphism graphs of vertex-primitive 2-arc-regular graphs were determined.

For each pair (G, Gα) in (7.5.1), denote Gα = H = K:L, where K = Zp for some prime p. Then H is a maximal subgroup of G or of G∆, where |G:G∆|=2, and G 6=

G∆ × Z2. Let g ∈ NG(L)\L be a 2-element. Then Γ = Cos(G, H, HgH) is G-vertex- primitive or G-vertex-biprimitive, (G, 2)-path-transitive but not (G, 2)-arc-transitive. The following two lemmas, Lemma 7.5.1 and Lemma 7.5.2, are essentially Lemma 3.4 and Lemma 3.5 [32], which are originally related to vertex-primitive 2-arc-regular graphs.

0 Lemma 7.5.1. Let G, Gα = H = K:L be as in (1). Then for any 2-elements g, g ∈ ∼ 0 σ 0 NG(L)\L, Cos(G, H, HgH) = Cos(G, H, Hg H) if and only if g ∈ g L for some σ ∈

NAut(G)(H) ∩ NAut(G)(L).

σ 0 Proof. Assume that g ∈ g L, where σ ∈ NAut(G)(H) ∩ NAut(G)(L). Then

Cos(G, H, HgH) =∼ Cos(Gσ,Hσ, (HgH)σ) = Cos(G, H, HgσH) = Cos(G, H, Hg0H).

Conversely, assume that Γ = Cos(G, H, HgH) =∼ Cos(G, H, Hg0H) = Σ. Let Ω = [G:H], and let ϕ be an isomorphism from Γ to Σ. Label the vertices H, Hg of Γ as α, β, respectively, and label the vertex Hg0 of Σ as β0. Since Γ is G-arc-transitive, we may assume that αϕ = α and βϕ = β0, so that Hϕ = H. As Γ =∼ Σ, we have (HgH)ϕ = Hg0H, and thus gϕ ∈ Hg0H. Therefore Gϕ = hH, giϕ = hH, g0i = G, that is, ϕ normalizes G. Since G has a trivial centralizer in S(Ω), ϕ can be viewed as an automorphism of G, so we may assume that ϕ ∈ Aut(G). Note that H ∩ Hg = L, so ϕ g ϕ g0 L = (H ∩ H ) = H ∩ H = L. Hence ϕ ∈ NAut(G)(H) ∩ NAut(G)(L). ϕ 0 ϕ 0 ϕh 0 Since g ∈ Hg H, assume that g = h1g h2, where hi ∈ H. Then g 1 = g h2h1. σ 0 Let σ = ϕh1 and h = h2h1 ∈ H. Then g = g h, and clearly σ ∈ NAut(G)(H).

Since h1 induces an automorphism of Γ , σ is an isomorphism from Γ to Σ. Note g g0 that the arc stabilizer Gαβ = H ∩ H and the arc stabilizer Gαβ0 = H ∩ H . Thus σ g σ g0 σ L = (H ∩ H ) = H ∩ H = L, so σ ∈ NG(L). Hence g ∈ NG(L), which implies that 0 −1 σ σ 0 0 h = (g ) g ∈ H ∩ NG(L) = NH (L) = L, so g = g h ∈ g L.

As in [32], we denote D = NAut(G)(H) ∩ NAut(G)(L), and Π = {g¯ = gL | g ∈

NG(L)\L}, where g is a 2-element. Then D has a natural action on Π deﬁned by

g¯d = g¯d, where d ∈ D, g¯ ∈ Π.

101 7.5. The numbers of the non-isomorphic graphs

Lemma 7.5.2. Let G, Gα = H = K:L be as in (1), and let n be the number of non- isomorphic graphs corresponding to G, H, L. Then

(a) n equals the number of the D-orbits on Π;

(b) If in addition G = Aut(G) and H is maximal in G, then n equals the number of

involutions in NG(L)/L.

0 ∼ Proof. For any two 2-elements g, g ∈ NG(L)\L, by Lemma 7.5.1, Cos(G, H, HgH) = Cos(G, H, Hg0H) if and only if gσ ∈ g0L for some σ ∈ D, and the latter holds if and only ifg ¯ and g¯0 are in the same D-orbit. Thus we obtain (a). For (b), note that each elementg ¯ ∈ Π is an involution of NG(L)/L. If G = Aut(G) and H is maximal in G, then D = NG(H) ∩ NG(L) = H ∩ NG(L) = NH (L) = L. Since L ﬁxes eachg ¯ ∈ Π, then n equals the number of involutions in NG(L)/L.

Lemma 7.5.3. Let (G, H) = (Sp, Zp:Z(p−1)/2), with p prime. Let H = K:L, where K = Zp, L = Z(p−1)/2. Then

(a) NG(H) = Zp:Zp−1;

(b) NG(L) = ((Z(p−1)/2 × Z(p−1)/2).2):Aut(Z(p−1)/2).

Proof. (a) Since H has a unique conjugacy class in Sp, so it is contained in a maximal ∼ subgroup M of Sp, where M = Zp:Zp−1. Note that H is a normal subgroup of M, so NG(H) = NM (H) = Zp:Zp−1. 2 (b) Denote NG(H) = K:M, where M = hti, with o(t) = p − 1. Let s = t . Then s is the product of two (p − 1)/2-cycles and L = hsi. Assume that Sp acts naturally on the set Ω = {1, 2, . . . , p} and t = (1, 2, . . . , p − 1). We consider C: = CG(L). Since L <

S(∆) = Sp−1 of G, where ∆ = {1, 2, . . . , p−1}, and Sp−1 is a maximal subgroup of G, so

C ≤ NG(L) < Sp−1. Thus C is contained in a maximal subgroup of Sp−1. Since hti ≤ C, C is transitive on ∆. Note that the normal subgroup L of C has two orbits on ∆, so we know that C is imprimitive on ∆. Thus C is contained in a maximal subgroup D which is isomorphic to S(p−1)/2 o2. It is easy to show that the centralizer of L in D is isomorphic ∼ to (Z(p−1)/2 × Z(p−1)/2).2 = C. Thus NG(L) ≤ ((Z(p−1)/2 × Z(p−1)/2).2):Aut(Z(p−1)/2).

Since Aut(Z(p−1)/2) < Sp, we conclude that the equality holds.

Now we are ready to determine the numbers of non-isomorphic graphs corresponding to the pairs in (1).

1) Assume that G = Ap, Gα = Zp:Z(p−1)/2 = K:L, with p prime, p ≡ 3(mod 4) and p 6= 7, 11, 23. Then by Lemma 7.5.3, NAut(G)(H) = Zp:Z(p−1), and thus D = Zp−1.

102 7.5. The numbers of the non-isomorphic graphs

Also by Lemma 7.5.3, NG(L) = (Z(p−1)/2 × Z(p−1)/2):Aut(Z(p−1)/2), so NG(L)/L = n1 n2 nm Z(p−1)/2:Aut(Z(p−1)/2). Let (p − 1)/2 = p1 p2 . . . pm , where pi are distinct prime si numbers. Then Aut( (p−1)/2) = Aut( n1 ) × · · · × Aut( nm ). Assume that 2 k(pi − 1). Z Zp1 Zpm Then a Sylow 2-subgroup S2 of Aut(Z(p−1)/2) is isomorphic to Z2s1 × · · · × Z2sm , and

all the involutions of NG(L)/L are in S2. It is easily shown that the number of the m involutions of S2 is 2 − 1. Note that D = Zp−1 = L × hσi, where o(σ) = 2. Further, D ﬁxes the involutionσ ¯ ∈ Π, and for any involutiong ¯ ∈ Π,g ¯ 6=σ ¯, the pair {g,¯ g¯σ} is a D-orbit. Thus the number of the D-orbits on Π is (2m − 2)/2 = 2m−1 − 1, that is, there m−1 are 2 − 1 non-isomorphic graphs corresponding to (G, Gα).

2) Assume that G = Th, B, or M, H = Zp:Z(p−1)/2 = K:L. Then by Lemma 7.5.2,

the number n of non-isomorphic graphs equals the number of involutions in NG(L)/L.

3) Assume that G = Sp, Gα = Zp:Z(p−1)/2 = K:L. Then by the discussion of Case 1 and Lemma 7.5.2, the number n of non-isomorphic graphs equals 2m − 1.

4) Assume that G = PΓU3(5), Gα = (Z7:Z3) × Z3 = K:L, where K = Z7, L = Z3 × Z3. Then by GAP, NG(L)/L contains a unique involution, and by Lemma 7.5.2, the graph is unique.

5) Assume that G = PGL3(4).hτi with o(τ) = 2, Gα = (Z7:Z3) × Z3 = K:L, where K = Z7, L = Z3 × Z3. Then by GAP, |NG(L)| = 54, and NG(L)/L contains a unique involution, so there is only one graph.

103 Chapter 8

Finite vertex-biprimitive edge-transitive tetravalent graphs

Recall that a graph Γ = (V,E) is called vertex-biprimitive if it is bipartite and its automorphism group acts transitively on the edge set E, preserves the two parts and acts primitively on each part. In [53] Ivanov and Iofinova classified vertex-biprimitive edge-transitive cubic graphs, based on the classification of amalgams of edge-transitive cubic graphs obtained by Goldschmidt [45]. As a natural generalization of Ivanov and Iofinova’s work and as a further application of our main result Theorem 4.1.1, in this chapter we classify vertex-biprimitive, edge-transitive graphs of valency 4. The main result of this chapter is Theorem 8.1.1. The layout of this Chapter is as follows. In Section 2 we present some of the graphs appearing in Theorem 8.1.1. In Section 3 we give some properties about the vertex stabilizers of edge-transitive tetravalent graphs. Then in Section 4, we give a reduction for the proof of Theorem 8.1.1 to the almost simple groups, and finally in Section 5 we deal with the almost simple type, and complete the proof of the main theorem.

8.1 The classiﬁcation

The classiﬁcation is given as follows.

Theorem 8.1.1. Let Γ be a connected bipartite graph of valency 4. Assume that Γ is G-biprimitive and G-edge-transitive, where G ≤ AutΓ . Assume further that G is intransitive on V . Then, for an edge {α, β}, one of the following holds:

2 (i)Γ = K4,4, and Z2:Z3 ≤ G < AutΓ = S4 o Z2.

(ii) |V | = 2p, 2p2, or 2p3, and one of the following holds, respectively,

G = Zp:Z4 with p > 5 and 4 | (p − 1), and AutΓ = G × Z2.

104 8.1. The classification

2 2 2 G = Zp:Z4 with 4 - (p − 1), or Zp:D8, and AutΓ = (Zp:D8) × Z2; 3 3 3 G = Zp:A4 or Zp:S4, and AutΓ = (Zp:S4) × Z2.

(iii) Γ is the standard double cover of a vertex-primitive arc-transitive graph.

∼ ∼ (iv) G = PSL2(p), where p is a prime with p ≡ ±13, ±37 (mod 40), Gα = Gβ = A4, p+ε and there are exactly 6 non-isomorphic graphs, where ε = 1 or −1 such that 3 divides p + ε.

f ∼ ∼ 3f−1−1 (v) G = PSL2(3 ) with f ≥ 3 prime, Gα = Gβ = A4, and Γ is one of 2 non- isomorphic graphs.

(vi) G is an almost simple group, there exists a normal subgroup K ¡ G such that K,

Kα, Kβ and AutΓ lie in TABLE D, and for each case there is a unique graph.

Moreover, if Γ is not arc-transitive, then G is one of the following groups: PSL2(p) f as in part (iv), PSL2(3 ) as in part (v), or PGL2(11), PSL2(13), PSL2(23), PGL3(7),

PGU3(5), PSp4(3), or Th, as in TABLE D. TABLE D

KKα Kβ AutΓ Comments

PGL2(11) D24 S4 PGL2(11)

PSL2(13) D12 A4 PGL2(13)

PSL2(23) D24 S4 PSL2(23)

PSL2(p)S4 S4 PGL2(p) p ≡ ±1 (mod 8) 2 2 PSL3(3) 3 :2S4 3 :2S4 PSL3(3).2 projective plane 2 2 PGL3(7) 3 :2A4 6 :S3 PGL3(7).2 2 2 PGU3(5) 3 :2A4 6 :S3 PGU3(5).2 3 3 PSp4(3) [3 ]:S4 [3 ]:2A4 PSp4(3).2 1+2 2 1+2 2 G2(3) (3+ × 3 ):2S4 (3+ × 3 ):2S4 G2(3).2 generalized hexagon 2 2 M12 3 :2S4 3 :2S4 M12.2 Weiss graph 9 2 7 Th [3 ].2S4 3 .[3 ].2S4 Th discovered in [21]

Remarks on Theorem 8.1.1:

(1) Unlike the cubic case, amalgams for tetravalent graphs are not known yet. The proof of Theorem 8.1.1 depends on the main result of Chapter 4.

(2) Finite vertex-primitive arc-transitive tetravalent graphs are given in [72], and so graphs in part (iii) of Theorem 8.1.1 are known.

(3) All the graphs in (iv) are the standard double covers of certain undirected or digraphs. Some of the graphs in (iv) are the standard double covers of the graphs

105 8.2. Examples

in (iii), while others are not, depending on the choice of Gβ (see Lemma 8.5.7 for details). The graphs in (v) are the standard double covers of certain digraphs.

(4) To our best knowledge, the PGL3(7)-graph and PGU3(5)-graph are new examples, and all other graphs appearing in Theorem 8.1.1 have been more or less known.

8.2 Examples

In this section, we construct and study the graphs appearing in Theorem 8.1.1. Let Γ = (V,E) be a connected bipartite G-edge-transitive graph of valency 4, and let {α, β} be an edge. Then the edge stabiliser Gαβ = Gα ∩ Gβ and Gα is transitive on Γ (α) with stabiliser (Gα)β = Gαβ, and Gβ acts transitively on Γ (β) with stabiliser

(Gβ)α = Gαβ. Thus, |Gα : Gαβ| = |Gβ : Gαβ| = 4. Since Γ is connected, it follows that hGα,Gβi = G. Conversely, for a group G and subgroups L, R < G such that L ∩ R is core-free, one can deﬁne a G-edge-transitive graph with vertex set V = [G : L] ∪ [G : R] such that {Lx, Ry} is an edge if and only if xy−1 ∈ LR. This graph is called a coset graph, and is denoted by Cos(G, L, R). It is clear that the following two results hold.

Lemma 8.2.1. Let G ≤ AutΓ be transitive on E and intransitive on V . Then for an edge {α, β}, Γ is isomorphic to Cos(G, Gα,Gβ).

g Lemma 8.2.2. The coset graphs Cos(G, L1,R1) = Cos(G, L2,R2) if and only if L2 = L1 g and R2 = R1, where g ∈ G.

Proof. Let α1, β1 be the vertices corresponding to L1,R1, respectively. Then the graph G Cos(G, L1,R1) has vertex set [G : L1] ∪ [G : R1] and edge set (α1, β1) . Let (α2, β2) be another edge, where α2 ∈ [G : L1] and β2 ∈ [G : R1]. Then the edge set of Cos(G, L1,R1) G also has the form (α2, β2) . g Let g ∈ G be such that (α1, β1) = (α2, β2), and let L2 = Gα2 and R2 = Gβ2 . Then g g L1 = L2 and R1 = R2, and thus Cos(G, L1,R1) can also be expressed as Cos(G, L2,R2), namely, Cos(G, L1,R1) = Cos(G, L2,R2).

Conversely, suppose that Γ := Cos(G, L1,R1) = Cos(G, L2,R2). Let αi, βi be the vertices corresponding to Li,Ri, respectively, where i = 1, 2. Then (α1, β1) and (α2, β2) g g are two edges, and hence there exists g ∈ G such that (α1, β1) = (α2, β2). So L1 = L2 g and R1 = R2.

Lemma 8.2.3. For a group G and subgroups L, R < G such that hL, Ri = G, L ∩ R is core-free, and |L : L ∩ R| = |R : L ∩ R| = 4, the coset graph Cos(G, L, R) is connected and G-edge-transitive of valency 4.

106 8.2. Examples

We note that Cos(G, L, R) is not G-vertex-transitive. Let G be a ﬁnite group, and let H be a core-free subgroup of G. Denote by [G : H] the set of right cosets of H in G, namely, [G : H] = {Hx | x ∈ G}. For an element g ∈ G, the coset graph of G with respect to H and g is the graph with vertex set [G : H], such that Hx and Hy are adjacent if and only if yx−1 ∈ HgH. This coset graph is G-vertex-transitive, and is denoted by Cos(G, H, HgH). For a (directed or undirected) graph Γ = (V,E), the standard double cover of Γ is deﬁned to be the undirected bipartite graph Γ˜ with parts V0 and V1, where Vi = {(v, i) | v ∈ V } such that two vertices (u, i) and (w, j) of Γ˜ are adjacent if and only if i 6= j and u, w are adjacent in Γ . The above two coset graph constructions are related in the following way.

Lemma 8.2.4. Let G be a group and L a subgroup. Assume that g ∈ G is such that L∩ Lg is core-free. Then Γ = Cos(G, L, Lg) is the standard double cover of Cos(G, L, LgL) 2 x g and Γ is arc-transitive if there exists x ∈ G such that x ∈ NG(L), and L = L .

The following examples are associated with certain almost simple groups. First we look at a few well-known graphs.

Example 8.2.5. Let G be one of the groups: PSL3(3), G2(3) or M12, and let H < G 2 1+2 2 2 be isomorphic to 3 :2S4, (3+ × 3 ):2S4 or 3 :2S4, respectively. Then the subgroups of G which are isomorphic to H are maximal and form two conjugacy classes. Let L, R < G be isomorphic to H and not be conjugate. Then hL, Ri = G, and both L and R contain Sylow 3-subgroups of G, and hence up to conjugacy we may assume that

L ∩ R contains a Sylow 3-subgroup Q say. It follows that L ∩ R = NL(Q) = NR(Q). Thus, |L : L ∩ R| = |R : L ∩ R| = 4, and so the graph Γ = Cos(G, L, R) is connected, G-vertex-biprimitive, and of valency 4.

The PSL3(3)-graph is the incidence graph of the projective plane PG(2, 3); the G2(3)-

graph is a generalized hexagon of order (3, 3); and the M12-graph was ﬁrst constructed by Weiss [126].

Example 8.2.6. Let G be one of the groups: PGL2(11), PGL2(13), or PSL2(23). Then ∼ ∼ G has two maximal subgroups L = S4 and R = D24. Notice that any Sylow 3-subgroup of G is isomorphic to Z3, so we may assume that both L and R contain a Sylow 3- subgroup Q. Then NL(Q) = S3, and NR(Q) = R. Thus L ∩ R = NL(Q) = S3, and

107 8.2. Examples

|L : L ∩ R| = |R : L ∩ R| = 4. Therefore the graph Γ = Cos(G, L, R) is G-vertex- biprimitive, G-edge-transitive, and of valency 4.

For G = PGL2(13), let T = PSL2(13). Then T ∩ L = A4 and T ∩ R = D12. As T ∩ L and T ∩ R are maximal subgroups of T , the graph Γ is also T -vertex-biprimitive.

3 Example 8.2.7. The group G = PSp4(3).2 has two maximal subgroups L = [3 ]:2S4 3 and R = [3 ]:(S4 × 2). Both L and R contain Sylow 3-subgroups of G. Thus, up to conjugacy, we may choose L, R such that L ∩ R contains a Sylow 3-subgroup Q say. It then follows that L ∩ R = NL(Q) = NR(Q), and hence |L : L ∩ R| = |R : L ∩ R| = 4. Therefore, the coset graph Γ = Cos(G, L, R) is G-vertex-biprimitive of valency 4. Let 3 3 T = soc(G) = PSp4(3). Then T ∩ L = [3 ]:2A4 and T ∩ R = [3 ]:S4. As T ∩ L and T ∩ R are maximal subgroups of T , the graph Γ is also T -vertex-biprimitive.

The next example presents a few inﬁnite families of graphs.

f Example 8.2.8. Let T = PSL2(q), where q = p with p prime. (1). Let q = p ≡ ±13, ±37 (mod 40), or q = 3f with f ≥ 3 prime. Then T contains a maximal subgroup H = A4. Let h ∈ H be of order 3. Then CT (h) = Z p+1 or Z p−1 . 2 2 g Let g ∈ CT (h) \ hhi. Then Cos(T,H,H ) is T -vertex-biprimitive of valency 4. (2). Let p ≡ ±1 (mod 8). Then T contains maximal subgroups that are isomorphic to S4 and form two conjugacy classes. Let L and R be two subgroups of T which are isomorphic to S4 and are not conjugate. It is easily shown that all subgroups of T of order 3 are conjugate. Hence, we may choose L and R such that L ∩ R = S3. Then Γ = Cos(T, L, R) is T -vertex-biprimitive of valency 4.

The subgroups L and R are conjugate in Aut(T ) = PGL2(p). It follows that Γ is vertex-transitive and AutΓ ≥ PGL2(p).

(3). Let G = PGL2(q), where p ≡ ±11, ±19 (mod 40). Then G contains maximal subgroups that are isomorphic to S4 and all of them are conjugate. Let H be such a subgroup, and let P < H be isomorphic to S3. Then NG(P ) = S3 × Z2. Thus there g exists an involution g ∈ NG(P )\H such that hH, gi = G and |H : H ∩ H | = 4. So g Cos(G, H, H ) is G-vertex-biprimitive of valency 4.

There are graphs associated with groups PGU3(5) and PGL3(7).

Example 8.2.9. Let G = PGU3(5).2 or PGL3(7).2. By the Atlas [19], G contains ∼ 2 ∼ 2 two maximal subgroups L = 3 :2S4 and R = 6 :D12. Both L and R contain Sylow 3- subgroups of G. Thus, by Sylow’s Theorem, there exists x ∈ G such that L∩Rx contains x 2 a Sylow 3-subgroup Q of G. It follows that L ∩ R = NL(Q) = 3 :(2 × S3). Hence |L : L ∩ Rx| = |Rx : L ∩ Rx| = 4, and the graph Γ = Cos(G, L, Rx) is G-vertex-biprimitive of valency 4. Let K = PGU3(5) < PGU3(5).2, or K = PGL3(7) < PGL3(7).2, respectively. 2 x 2 x Then K ∩ L = 3 :2A4 and K ∩ R = 6 :S3. Since both K ∩ L and K ∩ R are maximal subgroups of K, the graph Γ is K-vertex-biprimitive. We remark that Γ is soc(G)-edge-transitive but Γ is not soc(G)-vertex-biprimitive.

108 8.3. Local structure of edge-transitive tetravalent graphs

The graph displayed in the next example was ﬁrst given in [21, page 98].

Example 8.2.10. The Thompson group G = Th has two maximal subgroups L = 9 2 7 [3 ].2S4 and R = 3 .[3 ].2S4. Both L and R contain Sylow 3-subgroups of G. Then Sylow’s Theorem tells us that there exists x ∈ G such that L ∩ Rx contains a Sylow x 9 x x subgroup Q of G. Thus, L ∩ R = NL(Q) = [3 ]:(2 × 3). Then |L : L ∩ R | = |R : x x L ∩ R | = 4, and the graph Γ = Cos(G, L, R ) is vertex-biprimitive of valency 4.

Finally, we construct examples of type aﬃne.

Example 8.2.11. Let H be the dihedral group D8. Then H has a unique faithful 2 irreducible representation of degree 2 over the ﬁeld Fp, with p > 2 prime. Let G = Zp:H, and write H = hai:hbi. Obviously, b ∈ GL2(p) is not a scalar, and it follows that b centralises an element x of G of order p. Then H ∩ Hx = hbi, and so Cos(G, H, Hx) is G-vertex-biprimitive, and G-edge-transitive of valency 4.

Example 8.2.12. Let H = A4. Then H has a unique 3-dimensional faithful irreducible 3 2 representation over Fp for p > 2. Let G = Zp:H, and write H = Z2:hbi, where o(b) = 3. Obviously, b ∈ GL3(p) is not a scalar, and it follows that b centralises an element x of G of order p. Then H ∩ Hx = hbi, and so Cos(G, H, Hx) is G-vertex biprimitive, and G-edge transitive of valency 4.

8.3 Local structure of edge-transitive tetravalent graphs

In this section, we collect some facts about edge-transitive tetravalent graphs. Let Γ = (V,E) be a connected graph. As before, for a vertex α, denote by Γ (α) the neighborhood of α. For a group G ≤ AutΓ , the permutation group induced by Gα Γ(α) [1] acting on Γ (α) is denoted by Gα , and the kernel of the action is denoted by Gα . Γ(α) ∼ [1] [1] [1] [1] [1] Then Gα = Gα/Gα . For vertices α, β, . . . , Γ , let Gαβ...Γ = Gα ∩ Gβ ∩ · · · ∩ GΓ . Assume that Γ is a ﬁnite bipartite G-edge-transitive graph of valency 4. Let U and W be the two parts of Γ , and let α ∈ U and β ∈ W such that {α, β} is an edge. Since Γ is regular of valency 4, we have |Gα| = |Gβ| and |Gα : Gαβ| = |Gβ : Gαβ| = 4. It is clear that both Gα and Gβ are {2, 3}-groups (see also [31]). Further, we have the following

Lemma 8.3.1. Assume that Γ is a ﬁnite bipartite G-edge-transitive graph of valency 4. Then for an edge {α, β}, one of the following holds:

(a) both Gα and Gβ are 2-groups;

Γ(α) Γ(β) (b) each of Gα and Gβ is A4 or S4;

Γ(α) ∼ Γ(β) ∼ 2 (c) relabeling if necessary, Gα = A4 or S4, and Gβ = Z4, Z2, or D8.

109 8.3. Local structure of edge-transitive tetravalent graphs

Proof. Assume ﬁrst that Gα is a 2-group. Then (a) holds. Assume next that 3 | |Gα|. Γ(α) Γ(β) Notice that each of Gα and Gβ is isomorphic to a transitive subgroup of S4, so Γ(α) Γ(β) we have three possibilities: 1) each of Gα and Gβ is isomorphic to A4 or S4, in Γ(α) Γ(β) 2 this case (b) holds; 2) each of Gα and Gβ is isomorphic to Z4, Z2, or D8. It then Γ(α) Γ(β) follows that both Gα and Gβ are 2-groups, a contradiction; 3) one of Gα and Gβ is 2 isomorphic to A4 or S4, and the other is isomorphic to Z4, Z2, or D8, that is, (c) holds.

Γ(α) The next example shows that it may happen that the induced groups Gα and Γ(β) Gβ are totally diﬀerent. ∼ ∼ Example 8.3.2. Let X = S4 act on U = {α1, α2, α3, α4}, and let Y = A4 act on

W = {β1, β2, β3, β4}, where both actions are transitive. Let G = X ×Y , acting naturally on U ∪ W . Then G acts biprimitively on the complete bipartite graph Γ with parts U ∼ ∼ and W . Further, Gα1 = Xα1 × Y = S3 × A4, and Gβ1 = X × Yβ1 = S4 × Z3. Hence G = X × Y =∼ S × . So G[1] = S , G[1] = , and GΓ(α1) = A , GΓ(β1) = S . α1β1 α1 β1 3 Z3 α1 3 β1 Z3 α1 4 β1 4 Γ(α) Another example is Example 8.2.6, where Gα = D24, Gβ = S4, and Gα = D8, Γ(β) Gβ = S4.

Since the graph Γ is ﬁnite and connected, there exists a path of length n: α0, [1] α1, . . . , αn such that Gα0α1...αn = 1. Let α = α1, and β ∈ Γ (α) \{α1}. Then

[1] [1] [1] [1] 1 = Gαα1...αn ¡ Gαα1...αn−1 ¡ ··· ¡ Gαα1 ¡ Gα ¡ Gβα,

[1] ∼ Γ(α) Gβα/Gα = Gβα , and for each i, the factor group

[1] [1] ∼ [1] [1] [1] ∼ [1] Γ(αi+1) Γ(αi+1) Gαα1...αi /Gαα1...αi+1 = (Gαα1...αi Gi+1)/Gαi+1 = (Gαα1...αi ) ¡ (Gαiαi+1 ) .

Γ(α) ∼ Γ(β) ∼ Γ(α) ∼ [1] Suppose that Gα = Gβ = A4. Then Gαβ = Z3, and thus Gαα1...αi is a 3-group for each i. It then follows that Gβα is 3-group, leading to the next lemma. Γ(α) ∼ Γ(β) ∼ Lemma 8.3.3. Assume that Gα = Gβ = A4, where {α, β} is an edge. Then Gαβ 2 is a 3-group, and a Sylow 2-subgroup of Gα is Z2. [k] For an integer k ≥ 1, let Gα be the pointwise stabiliser of all vertices with distance at most k from α. The largest normal p-subgroup of X is denoted by Op(X). A graph Γ(α) Γ is called G-locally primitive if Gα is primitive for each vertex α. The following fundamental result is due to J. Van Bon, which is an extension of the well-known Thompson-Wielandt theorem on the structure of the point stabilizers of locally primitive graphs.

Theorem 8.3.4. ([5, Theorem 1.1]) Let Γ = (V,E) be a ﬁnite connected G-locally [1] primitive graph. Then for an edge {α, β}, there exists a prime p such that either Gαβ [1] [2] [2] [3] is a p-group, or Gαβ = Gα and Gβ = Gβ is a p-group.

110 8.4. A reduction

For tetravalent graphs, we have the following properties.

Lemma 8.3.5. Let Γ = (V,E) be a connected G-edge-transitive tetravalent graph. As- Γ(α) Γ(β) ∼ sume that, for an edge {α, β} ∈ E, Gα and Gβ are conjugate in G, and Gα ,Gβ = ∼ A4 or S4. Then either Gα = A4 or S4, or Gα is a 3-local group, and one of the following holds:

[1] [1] (i) Gα 6= 1, Gαβ = 1, and Z3 × A4 ≤ Gα ≤ S3 × S4.

[1] 5 (ii) 1 6= Gαβ is a 3-group, and 2 - |Gα|.

[1] [2] 8 (iii) 1 6= Gαβ is not a 3-group, Gβ is a 3-group, and 2 - |Gα|.

[1] ∼ Proof. If Gα = 1, then Gα acts faithfully on Γ (α) and Gα = S4. [1] [1] Thus, we next assume that Gα 6= 1. Assume further that Gαβ = 1. Then

[1] Γ(α) [1] [1] Γ(β) Γ(α) [1] Γ(β) Γ(α) Gα = Gα .Gα = (Gαβ.(Gα ) ).Gα = (Gα ) .Gα .

[1] Γ(β) Γ(β) [1] Γ(β) ∼ Since 1 6= (Gα ) ¢ Gαβ = Z3 or S3, we have (Gα ) = Z3 or S3. It follows that ∼ [1] Γ(β) Γ(α) ∼ [1] Γ(α) Γ(α) Γ(β) ∼ Gα = (Gα ) .Gα = Gα × Gα . Since Gα ,Gβ = A4 or S4, we conclude that part (i) holds. [1] [1] Suppose now that Gαβ 6= 1. Let N be a minimal normal subgroup of Gα . There exists a k-arc (α0 = α, α1, α2, . . . , αk) such that N ﬁxes Γ (αi) pointwise for i ≤ k − 1 d ∼ and does not ﬁx Γ (αk). Then Zq = N ≤ Gαk acts non-trivially on Γ (αk), and so

Γ(αk) Γ(αk) 1 6= N ¡ Gαk−1αk . It follows that q = 3. Further, each minimal normal subgroup [1] [1] of Gα is a 3-subgroup. Thus, soc(Gα ) is a normal subgroup of Gα, and hence Gα is a

3-local subgroup which implies that Gβ is also a 3-local. [1] [2] [1] [2] By Theorem 8.3.4, either Gαβ or Gβ is a p-group. Since Gαβ and Gβ are normal [1] in Gβ , they are 3-local. So p = 3. [1] [1] [1] [1] Γ(β) [1] Γ(β) If Gαβ is a 3-group, then as Gα = Gαβ.(Gα ) and (Gα ) ≤ S3, we conclude 5 that 2 - |Gα|, as in part (ii). [1] [2] [2] ∼ Γ2(β) Γ(α) Γ(β) If Gαβ is not a 3-group, then Gβ is a 3-group. As Gβ/Gβ = Gβ ≤ Gαβ oGβ , Γ(α) Γ(β) 8 Gαβ ≤ S3, and Gβ ≤ S4, we conclude that 2 - |Gβ| = |Gα|, as in part (iii).

8.4 A reduction

Let Γ = (V,E) be a connected bipartite graph of valency 4 with two parts U and W . Assume that G is transitive on E and intransitive on V , and assume further that G is primitive on both U and W . Suppose that G acts unfaithfully on U. Let K be the kernel of G acting on U. Since G is faithful on V Γ , the kernel K of G acts faithfully on W . Since K ¢ G and G is

111 8.4. A reduction primitive on W , we conclude that K is transitive on W . It follows that Γ is a complete 2 bipartite graph and |U| = |W | = 4. Thus, Z2 ≤ K ≤ S4, and we have the following conclusion.

2 Lemma 8.4.1. If G is unfaithful on U, then Γ = K4,4 and Z2:Z3 ≤ G < AutΓ = S4 oS2.

k From now on, assume that G is faithful on U. Let N = soc(G) = T1 × · · · × Tk = T , with k ≥ 1 and Ti = T simple. Then N is transitive on U and W .

Lemma 8.4.2. The socle N is abelian or simple.

Proof. Suppose that k > 1 and T is nonabelian. For α ∈ U, since 1 6= Nα ¢ Gα, it is readily seen Γ is a N-edge-transitive tetravalent graph. Let M = T1 × · · · × Tk−1. Then

M ¡ N. Since Nα is a {2, 3}-group, M is intransitive on U. Then the quotient graph

ΓM is N/M-edge-transitive and has valency dividing 4. Since T = N/M ≤ AutΓM , the valency of ΓM is 4, and hence M is semiregular on U. By the O’Nan-Scott Theorem (see [24]), G is not primitive on U, which is a contradiction. Thus, either k = 1 and N is simple, or N is abelian.

n Next let N = soc(G) = Zp , with p prime and n ≥ 1. Then N is a regular normal subgroup of G, and we may identify both U and W with N. Let α be the zero vector.

Then G = N:Gα and Gα is an irreducible subgroup of GLn(p). By [41], Gα is faithful 2 on Γ (α), and so Gα = Z4, Z2,D8,A4 or S4. It is known (see [104] for example) that the faithful irreducible representations of transitive permutation groups of degree 4 are the following:

(1) The cyclic group Z4 has a unique faithful irreducible representation over Fp for p > 2. The degree of the representation is 1 if p ≡ 1(mod 4), and is 2 if p ≡ 3(mod 4).

(2) The dihedral group D8 has a unique faithful irreducible representation of degree

2 over Fp for p > 2. For example, let

0 1 1 0 A = ,B = . −1 0 0 −1 ∼ Then D8 = hA, Bi.

(3) The alternating group A4 has a unique 3-dimensional faithful irreducible representation over Fp for p > 2. For example, let

 0 −1 0   1 0 0  A =  0 0 −1  ,B =  0 −1 0 . 1 0 0 0 0 −1 ∼ Then A4 = hA, Bi.

112 8.4. A reduction

(4) The symmetric group S4 has exactly two faithful irreducible representations of degree 3 over Fp for p > 2. Assume that S4 = {x, y, s, t}, where x = (14)(23), y = (13)(24), s = (243), t = (34). Using GAP, the two representations can be described as follows.

Let

 −1 0 0   1 0 0   0 0 1  A =  0 1 0  ,B =  0 −1 0  ,C =  1 0 0  , 0 0 −1 0 0 −1 0 1 0

 0 0  D =  0 0  , where = ±1. 0 0

Deﬁne the map φ which maps x, y, s, and t to A, B, C, and D, respectively. Then ∼ S4 = P = {A, B, C, D} are two faithful irreducible representations of S4. This gives the following lemma.

n Lemma 8.4.3. For soc(G) = Zp , the group G is one of the following groups:

2 2 3 3 Zp:Z4 with p ≡ 1(mod 4), Zp:Z4 with p ≡ 3(mod 4), Zp:D8, Zp:A4, or Zp:S4.

Proof. As discussed above, the group G is as in Theorem 8.1.1 (ii).

2 Assume that Gα = Z4 and G = Zp:Z4 or Zp:Z4. Then none of the non-identity

elements of Op(G) is centralised by a non-identity element of Gα. Thus, for an element x x x ∈ Op(Gα) of order p, the intersection Gα ∩ Gα = 1, and so the graph Cos(G, Gα,Gα) is vertex-biprimitive and edge-transitive of valency 4.

2 3 For G = Zp:D8 or G = Zp:A4, Examples 8.2.11 and 8.2.12 show the existence of vertex-biprimitive edge-transitive tetravalent graphs.

3 Finally, assume that Gα = S4 and G = Zp:S4. Let K < Gα be isomorphic to S3. It is x easily shown that K centralises a non-identity element x ∈ Op(G). Then Gα ∩ Gα = K, x and the coset graph Cos(G, Gα,Gα) is vertex-biprimitive and edge-transitive and of valency 4.

Remark: The graphs corresponding to the groups of Lemma 8.4.3 were constructed (by a diﬀerent approach) as bipartite graphs or the standard double covers of certain Cayley graphs in [40].

113 8.5. Almost simple type

8.5 Almost simple type

We treat the almost simple case in this section. Let Γ = (V,E) be a connected graph of valency 4. Let {α, β} be an edge of Γ . We ﬁrst prove a technical lemma which will be used frequently.

Lemma 8.5.1. Assume that Gα,Gβ are conjugate (in G), and the subgroups of Gα which are isomorphic to Gαβ are all conjugate (in Gα). Then there exists g ∈ NG(Gαβ) g such that Gβ = Gα. In particular, NG(Gαβ) 6⊂ Gα ∪ Gβ.

x Proof. Since Gα,Gβ are conjugate, there exists x ∈ G such that Gα = Gβ. Then ∼ x x xh x h Gαβ = Gαβ < Gα = Gβ. Thus, there exists h ∈ Gβ such that Gαβ = (Gαβ) = Gαβ. g xh h Now g := xh ∈ NG(Gαβ) is such that Gα = Gα = Gβ = Gβ.

Assume that Γ is bipartite with two parts U and W such that G ≤ AutΓ is transitive on E and intransitive on V . As noticed before, Gα and Gβ are {2, 3}-groups. Assume further that G is almost simple and primitive on both U and W . Then Gα and Gβ are maximal subgroups of G. Moreover, since Γ is of valency 4, the intersection Gαβ =

Gα ∩ Gβ has index 4 in both Gα and Gβ.

Notice that the pairs (G, H), with G almost simple and H = Gα or Gβ, are given in

TABLE I–TABLE VII of Chapter 4. Our task is to identify all the triples (G, Gα,Gβ).

We ﬁrst deal with the case where Gα is a 2-group.

Lemma 8.5.2. Let Gα be a 2-group. Then G = PGL2(7), PGL2(9), M10, PΓL(2, 9) or PSL(2, 17), and Γ is the standard double cover of an arc-transitive graph.

Proof. Since Gα,Gβ are both maximal 2-subgroups of G, it follows that Gα,Gβ are Sylow 2-subgroups of G, and hence they are conjugate. Therefore by [31, Lemma 2.5], Γ is the standard double cover of a vertex-primitive arc-transitive graph of valency 4 (directed or undirected). The insoluble primitive permutation groups which have a suborbit of length 4 are listed in [72, Theorem 3.4]. Since subgroups of G which are isomorphic to Gα are conjugate, the pair (G, Gα) must be in that list. Inspecting the list, we conclude that the pair (G, Gα) is one of the following:

G PGL2(7) PGL2(9) M10 PΓL(2, 9) PSL(2, 17) 5 Gα D16 D16 Z8:Z2 [2 ]D16

By [72, Theorem 1.5], each pair (G, Gα) corresponds to a vertex-primitive arc-transitive graph of valency 4, hence the result follows.

We thus assume that 3 divides |Gα| = |Gβ|.

114 8.5. Almost simple type

∼ Lemma 8.5.3. If Gα =6 Gβ, then (G, Gα,Gβ) lies in the following table:

G PGL2(11) PGL2(13) PSL2(23) PGL3(7) PSp4(3) PGU3(5) Th 2 3 2 9 Gα S4 S4 S4 3 :2A4 [3 ]:2A4 3 :2A4 [3 ].2S4 2 3 2 2 7 Gβ D24 D24 D24 6 :S3 [3 ]:S4 6 :S3 3 .[3 ].2S4

Moreover, there is a unique graph corresponding to each case.

∼ Proof. Since Gα =6 Gβ, the case (b) or (c) of Lemma 8.3.1 holds. Notice that |Gα| =

|Gβ|. Inspecting TABLE I–TABLE VII, we found that the candidates of the triples

(G, Gα,Gβ) are as follows:

(PGL2(11), D24, S4), (PSL2(13), D12, A4), (PGL2(13), D24, S4), (PSL2(23), D24, S4), 2 2 3 3 2 2 (PGL3(7), [3 ]:2A4, [3 ]:S4), (PSp4(3), [3 ]:2A4, [3 ]:S4), (PGU3(5), 3 :2A4, 6 :S3), 9 2 7 and (Th, [3 ].2S4, 3 .[3 ].2S4). By Examples 8.2.6, 8.2.7, 8.2.9 and 8.2.10, for each of the cases, there exists a G-vertex-biprimitive, G-edge-transitive tetravalent graph Γ . Further, as indicated in Examples 8.2.6, 8.2.7 and 8.2.9, those groups with the same socle correspond to the same graph.

By the Atlas [19], all subgroups of G that are isomorphic to Gα are conjugate, and

hence we can ﬁx Gα. Further, subgroups of Gα of index 4 are all conjugate, and thus

we may ﬁx such a subgroup K (as the possible arc stabiliser Gαβ). Then the candidates

for Gβ are subgroups R < G such that R 6= Gα and R > K. Let R1,R2 be two such x x subgroups. Then R1,R2 > K and R1 = R2 for some x ∈ G. Now K,K < R2, and xh xh h so K = K for some h ∈ R2. Thus, xh ∈ NG(K), and R1 = R2 = R2. Notice that K = Q:2 or Q:[4], where Q is a Sylow 3-subgroup of G. It follows from the information xh given in the Atlas [19] that NG(K) ≤ NG(Q) < R1. So R1 = R1 = R2, and hence there is only one corresponding graph.

∼ Lemma 8.5.4. If Gα = Gβ and Gα,Gβ are not conjugate in G, then Γ is vertex- transitive, and (G, Gα,Gβ) lies in the following table:

G M12 PSL3(3) G2(3) PSL2(p) with p ≡ ±1 (mod 8) ∼ 2 2 1+2 2 Gα = Gβ 3 :2S4 3 :2S4 (3+ × 3 ):2S4 S4

Moreover, in each case the corresponding graph is unique.

Proof. Assume that the maximal subgroups of G that are isomorphic to Gα form two conjugacy classes C1 and C2. By TABLE I, we ﬁnd that (G, Gα) is one of the 2 2 1+2 2 following pairs: (A6, S4), (M12, 3 :2S4), (PSL3(3), 3 :2S4) , (G2(3), (3+ × 3 ):2S4), or

(PSL2(p), S4), where p ≡ ±1 (mod 8).

115 8.5. Almost simple type

For (G, Gα,Gβ) = (A6, S4, S4) with Gα and Gβ being not conjugate, the intersection

Gα ∩ Gβ should be isomorphic to S3, which is not possible. For each of the other cases, the existence of Γ is shown in Example 8.2.5 or 8.2.8.

Suppose that Gα ∈ C1. Since all subgroups of Gα of index 4 are conjugate in Gα, we may choose such a subgroup K (as potential Gαβ). Then candidates for Gβ are subgroups R ∈ C2 which contains K. Arguing as in the proof of Lemma 8.5.3, we conclude that there is only one suitable graph.

Thus, we only need to treat the cases where Gα and Gβ are conjugate in G.

Lemma 8.5.5. Assume that all maximal subgroups of G that are isomorphic to Gα are conjugate (in G), and all subgroups K of Gα of index 4 are conjugate and maximal

(in Gα). Then there exist exactly |NG(K)/K| − 1 diﬀerent graphs which are G-edge- ∼ transitive and G-vertex-biprimitive of valency 4. In particular, if NG(K)/K = Z2, then there is a unique graph Cos(G, Gα,Gβ), which is the standard double cover of

Cos(G, Gα,GαgGα), where g ∈ NG(K) \ K.

Proof. Since maximal subgroups of G that are isomorphic to Gα are all conjugate, we can ﬁx Gα. Let K be a ﬁxed subgroup of Gα of index 4. By Lemma 8.5.1, any suitable g graph has the form Cos(G, Gα,Gα), for some element g ∈ NG(K) \ K. On the other g hand, each element g ∈ NG(K) \ K gives rise to a graph Γ = Cos(G, Gα,Gα), which is G-edge-transitive and G-vertex-biprimitive graph of valency 4.

g1 g2 For two elements g1, g2 ∈ NG(K) such that g2 ∈ g1K, we have Gα = Gα , and g1 g2 g1 g2 so Cos(G, Gα,Gα ) = Cos(G, Gα,Gα ). Conversely, assume that Gα = Gα , where −1 g1 g2 −1 g1, g2 ∈ NG(K). Then Gα = Gα. Since Gα is maximal in G, we have g1 g2 ∈ Gα, −1 and hence g1 g2 ∈ Gα ∩ NG(K) = NGα (K). Moreover, since K is maximal in Gα, we conclude that NGα (K) = K, and so g2 ∈ g1K. Therefore, there exist exactly |NG(K)/K| − 1 diﬀerent graphs.

Lemma 8.5.6. Assume that Gα,Gβ are conjugate and isomorphic to S4. Then G = A6, or G = PSL2(p), with p ≡ ±1 (mod 8), or G = PGL2(p) with p ≡ ±11, ±19 (mod 40), and Γ is the standard double cover of an arc-transitive graph.

Proof. By TABLE I–TABLE VII, we have (G, Gα) = (A6, S4), or (PSL2(p), S4) with p ≡ ±1 (mod 8), or (PGL2(p), S4), where p ≡ ±11, ±19 (mod 40). ∼ For each case, notice that Gα and Gβ are conjugate, Gαβ = S3, and all the subgroups of Gα that are isomorphic to Gαβ are conjugate. By Lemma 8.5.1, there ex- g ists g ∈ NG(Gαβ)\Gα such that Gβ = Gα. Furthermore, NG(Gαβ) = Gαβ × Z2 and NG(Gαβ)/Gαβ = Z2. By Lemma 8.5.5, there is a unique graph corresponding to this case, which is the standard double cover of a vertex-primitive, arc-transitive tetravalent graph.

116 8.5. Almost simple type

Lemma 8.5.7. Assume that Gα,Gβ are conjugate and isomorphic to A4. Then either

G = A5,Γ = K5,5 − 5K2, or G = PSL(2, q), and Γ is the standard double cover of an undirected graph or a digraph, where either

p+ (i) q = p ≡ ±13, ±37 (mod 40), and there are exactly 3 −1 diﬀerent suitable graphs; or

(ii) q = 3f with f ≥ 3 prime, and there are exactly 3f−1 − 1 diﬀerent graphs that are G-vertex-biprimitive and G-edge-transitive of valency 4.

Proof. By TABLE I–TABLE VII, the candidates for (G, Gα) are (A5, A4), and (PSL2(p), A4), f where q = p ≡ ±13, ±37 (mod 40), or q = 3 with f ≥ 3 prime. If (G, Gα) = (A5, A4), then Γ = K5,5 − 5K2, which is the standard double cover of the graph K5.

Thus, we consider the case where (G, Gα) = (PSL2(q), A4). Notice that Gα and ∼ Gβ are conjugate, Gαβ = Z3, and all subgroups of Gα that are isomorphic to Gαβ are conjugate and maximal in Gα. By Lemma 8.5.1, there exists g ∈ NG(Gαβ)\Gα such g that Gβ = Gα. ∼ Let q = p ≡ ±13, ±37 (mod 40). Then NG(Gαβ) = Dq+ε, where 3 divides q + ε. g g For each element g ∈ NG(Gαβ) \ Gαβ, the subgroup Gα is such that Cos(G, Gα,Gα) is G-vertex-biprimitive and G-edge-transitive of valency 4. Thus, by Lemma 8.5.5, there q+ε are exactly 3 − 1 diﬀerent graphs. f f For the case where q = 3 with f ≥ 3 prime, the normaliser NG(Gαβ) = Z3 is ∼ f−1 a Sylow 3-subgroup of G, and thus NG(Gαβ)/Gαβ = Z3 . By Lemma 8.5.5, there are exactly 3f−1 − 1 diﬀerent graphs of valency 4 which are G-vertex-biprimitive and G-edge-transitive.

In the rest of this section, we consider the case where Gα and Gβ are conjugate and [1] Gα 6= 1.

Lemma 8.5.8. If Gαβ contains a Sylow 3-subgroup Q of G, then NG(Q) 6⊂ Gα ∪ Gβ.

g g Proof. Let Gβ = Gα, where g ∈ G. Then Q, Q are two Sylow 3-subgroups of Gβ, and gh h gh so Q = Q for some element h ∈ Gβ. Thus gh ∈ NG(Q) and Gα 6= Gβ = Gβ = Gα . So NG(Q) 6⊂ Gα. Similarly, NG(Q) 6⊂ Gβ, and so NG(Q) 6⊂ Gα ∪ Gβ.

[1] Γ(α) Lemma 8.5.9. If 3 | |Gα|, and Gα 6= 1, then Gα is not a 2-group.

{2, 3}-subgroup Gα of G is such that Gα has a factor group isomorphic to A4 or S4, and has another factor group which is isomorphic to a transitive 2-subgroup of S4. Checking

TABLE I–TABLE VII, no such a pair (G, Gα) exists, a contradiction.

117 8.5. Almost simple type

Γ(α) ∼ Γ(β) ∼ It follows from Lemma 8.5.9 that Gα = Gβ = A4 or S4. Therefore by Lem- Γ(α) ∼ Γ(β) ∼ ma 8.3.5, Gα is 3-local. Assume that Gα = Gβ = A4, then by Lemma 8.3.3, a 2 Sylow 2-subgroups of Gα is Z2. This leads to the following [1] Lemma 8.5.10. Assume that Gα 6= 1. Then Gα satisﬁes the following properties:

8 (a) 2 - |Gα|;

(b) The Sylow 3-subgroups of Gα are not normal in Gα;

Γ(α) ∼ Γ(β) ∼ 2 (c) Either Gα = Gβ = A4, and a Sylow 2-subgroup of Gα is Z2, or Gα has a factor group which is isomorphic to S4.

Lemma 8.5.11. Assume that Gα and Gβ are conjugate in G and Gα is 3-local. Then one of the following holds:

(i) (G, Gα) = (PΣL2(27), A4 × 3), or (PSL2(27).6, S4 × 3).

(ii) (G, Gα) = (A7, (A4 × 3).2) or (S7, S4 × S3), and Γ is the standard double cover of

the odd graph O3.

(iii) (G, Gα) = (PSL3(7), (A4 × 3).2), and there is only one corresponding graph.

Proof. Let T = soc(G). We distinguish four cases.

Case 1. Let T be an alternating group. Then (G, Gα) lies in TABLE I and sat- isﬁes the conditions of Lemma 8.5.10. Inspecting TABLE I, the only possibilities are 3 3 4 3 (G, Gα) = (A7, (A4 × 3):2), (S7, S4 × S3), (A9, 3 :S4), (S9, 3 :(2 × S4), (A12, 3 :2 :S4), or

(S12, S3 o S4). 3 4 3 Suppose that (G, Gα) = (A9, 3 :S4) or (A12, 3 :2 :S4). Then subgroups of G which are isomorphic to Gα are all conjugate. Notice that |Gα:Gαβ| = 4 and Gαβ contains a

Sylow 3-subgroup, say Q, of G. By the Atlas [19], we have NG(Gαβ) ≤ NG(Q) < Gαβ, 3 which is a contradiction. Similarly, the cases (G, Gα) = (S9, 3 :(2 × S4)) or (S12, S3 o S4) are not possible. 2 Thus, (G, Gα) = (A7, (A4 × 3):2) or (S7, S4 × S3). For the former, Gαβ = Z3:Z2, and 2 2 by the Atlas [19], NG(Gαβ) = Z3:Z2, and NG(Gαβ)/Gαβ = Z2. By Lemma 8.5.5, Γ is the standard double cover of a vertex-primitive, arc-transitive tetravalent graph, which is actually the odd graph O3. It is then easily shown that the pair (G, Gα) = (S7, S4 ×S3) gives rise to the same graph.

Case 2. Let T = soc(G) be a sporadic group. Then the candidates for (G, Gα) are given in TABLE II. Combining the inspection of which with Lemma 8.5.10, we obtain the list as follows:

T M12 Th Th Co1 Suz B 2 9 2 7 3+4 2 2+4 2 11 Tα 3 :2S4 [3 ].2S4 3 .[3 ].2S4 3 :2S4 3 :2.(A4 × 2 ).2 [3 ].(S4 × 2S4)

118 8.5. Almost simple type

Suppose that G = M12, Th, or Suz. By the Atlas [19], the maximal subgroups of G

that are isomorphic to Gα are all conjugate, and Gα contains a Sylow 3-subgroup Q of G.

It is easily shown that NG(Q) < Gα, which is not possible by Lemma 8.5.8. Analogously,

the cases G = Aut(M12) = M12.2 or G = Aut(Suz) = Suz.3 are not possible. 3+4 2 Γ(α) ∼ [1] Suppose that (G, Gα) = (Co1, 3 :2S4). Then Gα = S4, so we have Gα = 3+4 3+4 [1] ∼ [1] ∼ 3 :2S4 and Gαβ = 3 :(2S3 ◦ 2S4). Thus, Gαβ/Gα = S3, and it follows that Gαβ = 3+4 [1] Γ(Γ) 3 :Q8. Let Γ ∈ Γ (β) \{α}. Then (Gαβ) should be S3. However, the group 3+4 3 :Q8 does not have a factor group that is isomorphic to S3, which is a contradiction. 11 Finally, suppose that (G, Gα) = (B, [3 ].(S4 × 2S4)). Then Gαβ contains a Sylow

3-subgroup Q of G, and NGαβ (Q) = Q:[8]. By the Atlas [19], NG(Q) is contained in 1+8 1+6 11 a maximal subgroup that is isomorphic to Fi23, 3 :2 .U4(2).2 or [3 ]:(S4 × 2S4).

If NG(Q) < Fi23, then inspecting the maximal subgroups of Fi23 given in the Atlas 10 [19], we have NG(Q) < [3 ].(2 × PSL3(3)), and hence NG(Q) = Q:[8] < Gαβ, which 1+8 1+6 is a contradiction. If NG(Q) < 3 :2 .U4(2).2, then NG(Q) = Q:[4], which is not 11 possible. If NG(Q) < [3 ]:(S4 ×2S4), then NG(Q) < Gα, which is again a contradiction Case 3. Let T = soc(G) be an exceptional group of Lie type. Then by the TABLE VII and Lemma 8.5.10, there are the following candidates: 1+2 2 3 1+2 2 0 4 (T,Tα) = (G2(3), (3+ × 3 ):2S4), ( D4(2), 3 :2S4), or ( F4(8) , 3 :2S4).

In each case, Tα contains a Sylow 3-subgroup Q of T and Tαβ = Q:[4]. It is easily shown that NT (Q) = Tαβ, which is a contradiction by Lemma 8.5.8. Case 4. Let T = soc(G) be a classical group. Then from the TABLE III–VI, the candidates are (G, Gα) = (PΣL2(27), A4 × 3) or (PSL2(27).6, S4 × 3), and (T,Tα), which are listed in the following table:

T Tα 2 PSL3(3) 3 :2S4 PSL3(7) (A4 × 3):2 4 2 PSL4(3) [3 ]:2.A4.2 1+4 PSL4(3).2 3+ :(2S4 × 2) 3 PSp4(3) [3 ]:S4 3+4 PSp6(3) 3 .2.(S4 × A4) 1+4 PSU4(3) 3+ :2S4 5 2 PSU6(2) [3 ].Q8:S3 7 PΩ7(3) [3 ].(2A4 × A4).2 + 1+4 PΩ8 (2).3 3+ :2S4

The cases (G, Gα) = (PΣL2(27), A4 × 3) or (PSL2(27).6, S4 × 3) give rise to a G- vertex-biprimitive graph in Lemma 8.5.7 (ii). ∼ For (G, Gα) = (PSL3(7), (A4 × 3):2), we have Gαβ = S3 × Z3, and by the Atlas 2 ∼ [19], we have NG(Gαβ) = 3 :Q8. Thus, NG(Gαβ)/Gαβ = Z2, and by Lemma 8.5.5,

119 8.6. Proof of Theorem 8.1.1

g Cos(G, Gα,Gα) is a unique graph, where g ∈ NG(Gαβ)\Gα. For the other cases, we will show that no suitable graph exists. 2 1+4 3 Assume that (G, Gα) = (PSL3(3), 3 :2S4), (PSL4(3).2, 3+ :(2S4×2)), (PSp4(3), [3 ]:S4), 1+4 + 1+4 (PSU4(3), 3+ :2S4), or (PΩ8 (2).3, 3+ :2S4). Then maximal subgroups of G that are isomorphic to Gα are all conjugate, and Gα contains a Sylow 3-subgroup Q of G. By the Atlas [19], we conclude that NG(Q) < Gα, which is not possible by Lemma 8.5.8. 3+4 Assume that (G, Gα) = (PSp6(3), 3 .2.(S4 × A4)). Then Gαβ contains a Sylow 3- subgroup Q of G, and Gαβ = Q:[4]. By the Atlas [19], NG(Q) is contained in a maximal 1+4 6 3+4 subgroup that is isomorphic to 3+ :2U4(2), 3 :PSL3(3), or 3 :2(S4 × A4). It follows that NG(Q) = Q:[4] and thus NG(Q) = Gαβ, contradicting Lemma 8.5.8. 4 2 7 Similarly, the cases (G, Gα) = (PSL4(3), [3 ]:2.A4.2) and (G, Gα) = (PΩ7(3), [3 ].(2A4×

A4).2) are not possible either. 5 2 Assume that (G, Gα) = (PSU6(2), [3 ].Q8:S3). Then Gαβ contains a Sylow 3-subgroup, say Q, of G. By the Atlas [19], NG(Q) is contained in a maximal subgroup that is i- 1+4 7 somorphic to U4(3).2 or 3 .(2 .3). It then follows that NG(Q) ≤ Gαβ, which is a contradiction by Lemma 8.5.8.

8.6 Proof of Theorem 8.1.1

In this ﬁnal section, we will complete the proof of the main theorem. All vertex-biprimitive edge-transitive graphs of valency 4 are listed in Theorem 8.1.1, shown in the last two sections. Our task here is to determine their automorphism groups and isomorphism classes. We ﬁrst prove a simple lemma.

Lemma 8.6.1. Let Γ = (V,E) be a connected bipartite graph with two parts U and W . Then V = U ∪ W is the only bipartition of V .

Proof. Suppose that V = U 0 ∪ W 0 is another bipartition. Then

V = (U ∩ U 0) ∪ (U ∩ W 0) ∪ (W ∩ U 0) ∪ (W ∩ W 0) is a partition of V . It follows that no vertex in (U ∩ W 0) ∪ (W ∩ U 0) is adjacent to a 0 0 0 vertex in (U ∩ U ) ∪ (U ∩ W ), which is a contradiction since Γ is connected.

Let Γ = (V,E) be a connected bipartite graph of valency 4 with parts U and W . Assume that G is transitive on E and intransitive on V , and assume further that G is ∼ U ∼ W primitive on U and W . Then G = GU = GW and G = G = G . Let A = AutΓ . By + Lemma 8.6.1, A preserves the bipartition V = U ∪ W . Let A = AU = AW . Then for + + + + an edge {α, β}, we have G ≤ A , Gα ≤ Aα and Gβ ≤ Aβ . Therefore, if A is faithful + + + on U (and so on W ), then (A ,Aα ,Aβ ) is one of the triples (G, Gα,Gβ) satisfying

120 8.6. Proof of Theorem 8.1.1

Lemmas 8.4.3, 8.5.2–8.5.4, 8.5.6–8.5.7, and 8.5.11. An inspection of these triples leads to the following lemma.

Lemma 8.6.2. If A+ is faithful on U, then either

+ (1) G = Z5:Z4, A = S5, and Γ = K5,5 − 5K2, or

(2) G is normal in A.

Proof. Suppose that G is soluble and A+ is insoluble. It follows from Lemmas 8.5.2– + 8.5.4, 8.5.6–8.5.7, and 8.5.11 that A = S5, and G = Z5:Z4 and Γ = K5,5 − 5K2, as in part (1). For the other cases, by Lemmas 8.4.3, 8.5.2–8.5.4, 8.5.6–8.5.7, and 8.5.11, we conclude that G is normal in A.

We next determine A = AutΓ .

Lemma 8.6.3. Assume that soc(G) is abelian. Then Γ is arc-transitive, and part (i) or part (ii) of Theorem 8.1.1 holds.

Proof. Let R = soc(G). Then G = R:Gα. By Lemma 8.4.3, we have R = Zp, with 2 3 Gα = Z4; or R = Zp, with Gα = Z4 or D8; or R = Zp, with Gα = A4 or S4. + + Let A = AutΓ , and let A = AU = AW . If A is unfaithful on U, then by

Lemma 8.4.1, we conclude that A = S4 o S2, as in part (i) of Theorem 8.1.1. Thus, we assume that A+ is faithful on U. If A+ is insoluble, by Lemma 8.6.2, part (i) of Theorem 8.1.1 holds. We hence further assume that A+ is soluble. By 2 3 + Lemma 8.4.3, if G = Zp:Z4 with p > 5, or G = Zp:D8, or Zp:S4, then A = G. Next 2 3 consider the cases where G = Zp:Z4 and G = Zp:A4. 2 First, let G = Zp:Z4 < AGL(2, p) with p ≡ 3 (mod 4). Then there exists g ∈ G g 2 of order p such that Γ = Cos(G, Gα,Gα). The group G has an overgroup X = Zp:D8 2 in AGL(2, p). Write Xα = Gα:hbi. Then b centralizes an element x ∈ Zp \{1}. Let ρ ∈ GL(2, p) be such that bρ = b and g = xρ. Then bg = gb. Let Y = Xρ. Then g g g + 2 Yα ∩ Yα = hbi, and hence Cos(G, Gα,Gα) = Cos(Y,Yα,Yα ). So A = Zp:D8. 3 2 Next let G = Zp:A4 and Gα = Z2:hzi. Then z centralizes a unique cyclic subgroup g 3 hgi of order p, and Γ = Cos(G, Gα,Gα). In AGL(3, p), there is an overgroup X = Zp:S4 2 of G. Then Xα = Gα:hbi = Z2:hz, bi, and hgi is the unique cyclic subgroup of G of order g g p which is centralised by hz, bi. Thus, Xα ∩ Xα = hz, bi, and hence Cos(G, Gα,Gα) = g + 3 Cos(X,Xα,Xα). So A = Zp:S4. + Finally, by [53], Γ is arc-transitive, and hence A = A .Z2. It follows that one of the following cases occurs:

+ ∼ G = Zp:Z4, and A = A .Z2 = (Zp:Z4).Z2 = (Zp:Z4) × Z2.

121 8.6. Proof of Theorem 8.1.1

2 2 + 2 ∼ 2 G = Zp:Z4 or Zp:D8, and A = A .Z2 = (Zp:D8).Z2 = (Zp:D8) × Z2.

3 3 + 3 ∼ 3 G = Zp:A4 or Zp:S4, and A = A .Z2 = (Zp:S4).Z2 = (Zp:S4) × Z2.

Each of them satisﬁes part (ii) of Theorem 8.1.1.

Next we determine the isomorphism classes of Γ for the almost simple case. Assume that G is almost simple. By Lemmas 8.5.2–8.5.4, 8.5.6–8.5.7, and 8.5.11, either Γ is ∼ ∼ unique, or G = PSL(2, q) and Gα = Gβ = A4.

Lemma 8.6.4. Let G = PSL(2, q), and Gα,Gβ be conjugate and isomorphic to A4, as in Lemma 8.5.7. Then the following hold:

p+ε (i) For q = p ≡ ±13, ±37 (mod 40), there are exactly 6 non-isomorphic graphs that are G-vertex-biprimitive and G-edge-transitive, where ε = 1 or −1 such that 3 | (p + ε);

(ii) For q = 3f with f ≥ 3 prime, there are exactly (3f−1 − 1)/2 non-isomorphic suitable graphs of valency 4.

Proof. Since all subgroups of G that are isomorphic to A4 are conjugate, we may ﬁx

Gα. Let K be a Sylow 3-subgroup of Gα. By Lemma 8.5.7, there are exactly n diﬀerent graphs:

gi Γi = Cos(G, Gα,Gα ),

p+ε f−1 with gi ∈ NG(K) such that giK 6= gjK if 1 ≤ i < j < n. Here n = 3 − 1 or 3 − 1, corresponding to q = p ≡ ±13, ±37 (mod 40), or q = 3f , respectively. ∼ Assume that Γi = Γj, where i < j. Let σ be an isomorphism between Γi and Γj. σ σ Then G ≤ (AutΓi) = AutΓj. By Lemma 8.6.2, AutΓj is almost simple and contains a σ σ normal subgroup G. It follows that G is normal in AutΓj, and hence G = G. Thus, σ induces an automorphism of G, namely,

σ gi σ σ gj Cos(G, Gα, (Gα ) ) = Γi = Γj = Cos(G, Gα,Gα ).

σ g gi σ gj g −1 By Lemma 8.2.2, Gα = Gα and (Gα ) = (Gα ) , where g ∈ G. Write τ = σg . Then gi gj τ ∈ NAut(G)(Gα) = S4 and τ maps Gα to Gα . It follows that τ is an involution and normalises a Sylow 3-subgroup of Gα. Without loss of generality, we may assume that

τ normalises K. Hence τ ∈ NAut(G)(K) = NG(K):hτi. First, consider the case where q = p ≡ ±13, ±37 (mod 40). We observe that the elements of NAut(G)(K) = D2(p+ε) that are centralised by τ lie in the center Z(NAut(G)(K)) =

Z2. It follows that τ centralises one involution of NG(K)/K and divides the other p+ε p+ε−6 3 − 2 non-identity elements of NG(K)/K into two orbits of size 6 . Therefore, in p+ε−6 p+ε NG(K)/K, there are exactly 6 + 1 = 6 images that are not conjugate under τ.

122 8.6. Proof of Theorem 8.1.1

f f For q = 3 with f ≥ 3 prime, the normaliser NG(K) = Z3 is a Sylow 3-subgroup ∼ f−1 f of G and NG(K)/K = Z3 . Note that NAut(G)(K) = NG(K):hτi = Z3 :Z2 such that 3f−1−1 τ inverts all elements of NG(K). It follows that there are exactly 2 non-identity 3f−1−1 elements of NG(K)/K that are not conjugate under τ. So there are exactly 2 non-isomorphic graphs.

Now we are ready to prove Theorem 8.1.1.

Proof of Theorem 8.1.1

If G is unfaithful on U, then Lemma 8.4.1 shows that Γ = K4,4, as in part (i) of Theorem 8.1.1. Assume that G is faithful on both U and W . By Lemma 8.4.2, the socle soc(G) is abelian or simple. Moreover, if soc(G) is abelian, then Lemma 8.6.3 shows that Γ is as in part (i) or (ii) of Theorem 8.1.1. We may thus assume that soc(G) is nonabelian simple, namely, G is almost simple.

Let Γ = Cos(G, Gα,Gβ), where {α, β} is an edge. If both Gα and Gβ are 2-groups, then Lemma 8.5.2 tells that Γ is the standard double cover of a vertex-primitive arc-transitive graph of valency 4, as in part (iii) of Theorem 8.1.1.

Suppose next that Gα and Gβ are not 2-groups. If in addition Gα and Gβ are not

conjugate in G, then by Lemmas 8.5.3 and 8.5.4, the triple (G, Gα,Gβ) and Γ satisfy part (vi) of Theorem 8.1.1.

The remaining case is that Gα and Gβ are conjugate. Then either Gα and Gβ are

A4 or S4, or Gα is a 3-local group. ∼ If Gα = S4, then Lemma 8.5.6 shows that G = PSL2(p) with p ≡ ±1 (mod 8) or

G = PGL2(p), with p ≡ ±11, ±19 (mod 40), and Γ is the standard double cover of a vertex-primitive arc-transitive graph of valency 4, as in part (iii) of Theorem 8.1.1. ∼ If Gα = A4, then by Lemmas 8.5.7 and Lemmas 8.6.4, the statements of part (iv) and part (v) of Theorem 8.1.1 are true.

Finally, if Gα is 3-local, then Lemma 8.5.11 shows that (G, Gα) = (A7, (A4 × 3)2)

or (PSL3(7), (A4 × 3).2), and Γ is the standard double cover of a vertex-primitive arc- transitive graph of valency 4, or (G, Gα) = (PΣL2(27), A4 × 3) or (PSL2(27).6, S4 × 3), and Γ is a G-vertex-biprimitive graph given in Lemma 8.5.7, (ii), as in part (iii) or (V) of Theorem 8.1.1.

123 Chapter 9

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