Isomorphic Cayley Graphs on Non-Isomorphic Groups)

UNIVERZA NA PRIMORSKEM FAKULTETA ZA MATEMATIKO, NARAVOSLOVJE IN INFORMACIJSKE TEHNOLOGIJE

Matematiˇcneznanosti Studijskiˇ program 2. stopnje

Mary Agnes SERVATIUS Izomorfni Cayleyevi graﬁ nad neizomorfnimi grupami (Isomorphic Cayley Graphs on Non-Isomorphic Groups)

Magistrsko delo

(Master’s Thesis)

Mentor: izr. prof. dr. Istv´anKov´acs Koper, 2014 Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 ii Kljuˇcnadokumentacijska informacija

Ime in PRIIMEK: Mary Agnes SERVATIUS Naslov magistrske naloge: Izomorfni Cayleyevi grafi nad neizomorfnimi grupami Kraj: Koper Leto: 2014 Steviloˇ listov: 56 Steviloˇ slik: 27 Steviloˇ referenc: 18 Mentor: izr. prof. dr. IstvánKovács UDK: Kljuˇcnebesede: Cayleyjev graf, cikliˇcengraf, isomorfnost grafov, grupa avtomorfizmov grafov, leksikografiˇcenprodukt grafov, konˇcnaabelova grupa, permutacijska grupa, venˇcniprodukt permutacijskih grup, loˇcnotranzitiven graf. Math. Subj. Class. (2010): 05C25 20B25 20B25 05C60

Izvleˇcek

V magistrskem delu obravnavamo Cayleyeve grafe in grupe. Poseben poudarek je namenjen problemu Cayleyevih izomorﬁzmov. Poglavje 1 vsebuje uvod v glavni izrek, ki poda potrebni in zadostni pogoj za izomorfnost dveh Cayleyevih digrafov

X1 = cay(G1,S1) in X2 = cay(G2,S2), kjer sta G1 in G2 neizomorfni abelski 2- grupi, digrafa X1 in X2 pa imata regularno cikliˇcnogrupo avtomorﬁzmov. Omenjeni rezultat je razˇsiritevrezultata Morrisove (glej J. Graph Theory 3 (1999), 345–362) v zvezi s p-grupami Gi, kjer je p liho praˇstevilo. Poglavje 2 vsebuje osnovne deﬁnicije in potrebno predpripravo za kasnejˇsapoglavja. V 3. poglavju so obravnavani izbrani primeri. V 4. poglavju je podan dokaz glavnega izreka. Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 iii

Key words documentation

Name and SURNAME: Mary Agnes SERVATIUS Title of Masters degree: Isomorphic Cayley Graphs on Non-isomorphic Groups Place: Koper Year: 2014 Number of pages: 56 Number of figures: 27 Number of references: 18 Mentor: Assoc. Prof. IstvánKovács,PhD UDK: Key words: Cayley graph, cyclic graph, isomorphic graphs, automorphism groups of graphs, lexicographic products of graphs, elementary abelian groups, permutation groups, wreath product, arc transitive graph Math. Subj. Class. (2010): 05C25 20B25 20B25 05C60

Abstract

In this thesis we explore Cayley graphs and groups, in particular, we are concerned with the Cayley isomorphism problem. Chapter 1 contains a high level introduction to the main theorem, in which a necessary and suﬃcient condition is given for two Cayley digraphs X1 = Cay(G1,S1) and X2 = Cay(G2,S2) to be isomorphic, where G1 and G2 are non-isomorphic abelian 2-groups, and the digraphs X1 and X2 have a regular cyclic group of automorphisms. This result extends that of Morris (see J. Graph Theory 3 (1999),

345–362) concerning p-groups Gi, where p is an odd prime. Chapter 2 introduces some preliminaries, and Chapter 3 provides a selection of examples. Chapter 4 contains a proof of the main theorem. Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 iv

Contents

1 Introduction 1

2 Preliminaries 4 2.1 Graphs ...... 4 2.1.1 Basic deﬁnitions ...... 4 2.2 Groups ...... 6 2.2.1 Basic deﬁnitions ...... 6 2.2.2 Automorphism groups ...... 8 2.2.3 Permutation groups and group actions ...... 11 2.2.4 Imprimitivity block systems ...... 13 2.2.5 Wreath products of groups ...... 14 2.3 Cayley graphs ...... 17 2.3.1 Arc transitive Cayley digraphs on Cyclic groups ...... 19 2.3.2 CI-Graphs ...... 19

3 Non-isomorphic groups with isomorphic Cayley graphs. 22 3.1 Inﬁnite graphs and free groups ...... 23 3.2 Graphs of the Platonic solids ...... 27

4 Proof of the Main Theorem 34 4.1 W-subgroups of abelian groups ...... 34 4.2 Proof of Theorem 4.1.4 ...... 37

5 Conclusions 48 Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 v

Bibliography 49 Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 vi

Acknowledgements

Thanks are due to Dragan Marusiˇcfor this opportunity, to my mentor IstvánKovács for his patience, and to my father, Herman Servatius, who taught me everything I know. Thank you Papa, for letting me peek at the vast explored and unexplored territories of mathematics, for going on random walks with me on objects in space, and most of all for distracting me in my most difficult times with your apparently endless supply of mathematical treats. It does not escape my notice that the time, energy and money spent on this project vastly outweigh its value. Many years ago, when I was much too young and naive to be suspicious of a grand promise, I signed a contract which effectively forced me to work for years at a job to which I was completely unqualified for less than nothing. My anger over this situation is the only thing which shields me from my grief and humiliation. I am not sure if I will ever completely recover. Work on this thesis was generously supported by grant TI508-7529165 from the Trowbridge Institute, Massachusetts, USA. My maiden name is Mary Agnes Franziska Sophie Servatius, and my married name is Mary Milaniˇc. The name on the cover of this thesis does not reflect the legal status of my name in the countries of which I am a citizen, although it is being used to refer to me. Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 1

Chapter 1

Introduction

A necessary and sufficient condition is given for two Cayley digraphs X1 = Cay(G1,S1) and X2 = Cay(G2,S2) to be isomorphic, where the groups Gi are nonisomorphic abelian 2-groups, and the digraphs Xi have a regular cyclic group of automorphisms. Our result extends that of Morris (see J. Graph Theory 3 (1999), 345–362) concerning p-groups Gi, where p is an odd prime. Let G be a finite group with identity element e.A Cayley digraph Cay(G, S) on G generated by a connection set S ⊆ G, e∈ / S, is a digraph with the vertex set G and the arc set {(x, y) | x−1y ∈ S}. In this paper we consider the problem of finding efficient necessary and sufficient conditions for two Cayley digraphs Cay(G1,S1) and

Cay(G2,S2) to be isomorphic, such that the groups Gi are nonisomorphic. Here we restrict to the special case that Gi are p-groups. For an overview on the general problem we refer to the survey paper by Li [12].

2 The simplest case is that the groups Gi have order p , and this has been charac- n terized by Joseph in [10]. Joseph’s result was generalized to |Gi| = p by Morris in [16] as follows.

Recall ﬁrst that, for two digraphs X1 and X2, the wreath product X1 o X2 is the digraph with vertex set V (X1) × V (X2), and an arc from (x1, x2) to (y1, y2) if and only if (x1, y1) is an arc of X1, or x1 = y1 and (x2, y2) is an arc of X2. We remark that the wreath product Aut(X1) o Aut(X2) acts as a group of automorphisms of

X1 o X2, explaining also the term wreath product. Here we follow the convention that Aut(X1) o Aut(X2) has active group Aut(X1) and passive group Aut(X2), i.e., Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 2

n ordered by the relation ≤po, where for two abelian groups G and H of order p ,

G ≤po H if and only if there is chain

H1 < H2 < ··· < Hm = H of subgroups of H such that

1. H1,H2/H1,...,Hm/Hm−1 are all cyclic, and

∼ 2. G = H1 × H2/H1 × · · · × Hm/Hm−1.

The main result of Morris is the following (see [16, Theorem 1.1]).

Theorem 1.1.1. Let X = Cay(G, S) be a Cayley digraph on an abelian group G of order pn, where p is an odd prime. Then the following are equivalent:

(1) The digraph X is isomorphic to a Cayley digraph on both Zpn and H, where H is an abelian group with |H| = pn.

(2) There exists a chain of subgroups G1 < ··· < Gm−1 in G such that

(i) G1, G2/G1, . . . , G/Gm−1 are cyclic groups,

(ii) G1 × G2/G1 × · · · × G/Gm−1 ≤po H,

(iii) for each 1 ≤ i ≤ m − 1, S \ Gi is a union of Gi-cosets.

(3) There exist Cayley digraphs X1,...,Xm on cyclic p-groups H1,...,Hm such ∼ that H1 × · · · × Hm ≤po H and X = Xm o · · · o X1.

Moreover, Theorem 1.1.1 with the assumption G = H was shown to be true for any (not necessarily abelian) group G of order pn (see [16, Theorem 6.1]), as well as when p = 2 and H is an elementary abelian group of order 2n (see [16, Theorem 5.1]). Our goal in this thesis is to extend Theorem 1.1.1 to the case when p = 2, that is, to show the following theorem. Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 3

Theorem 1.1.2. Let X = Cay(G, S) be a Cayley digraph on an abelian group G of order 2n. Then the following are equivalent:

(1) The digraph X is isomorphic to a Cayley digraph on both Z2n and H, where H is an abelian group with |H| = 2n.

(2) There exists a chain of subgroups G1 < ··· < Gm−1 in G such that

(i) G1, G2/G1, . . . , G/Gm−1 are cyclic groups,

(ii) G1 × G2/G1 × · · · × G/Gm−1 ≤po H,

(iii) for each 1 ≤ i ≤ m − 1, S \ Gi is a union of Gi-cosets.

(3) There exist Cayley digraphs X1,...,Xm on cyclic 2-groups H1,...,Hm such ∼ that H1 × · · · × Hm ≤po H and X = Xm o · · · o X1.

Notice that the aforementioned nonabelian version of Theorem 1.1.1 does not hold

n for p = 2. Consider the cycle C2n on 2 vertices. Although the graph C2n is not decomposed nontrivially into wreath products of smaller graphs, it admits a regular group of automorphisms isomorphic to Z2n , and another one isomorphic to the dihedral group D2n . We remark that our result has been published in: J. Graph Theory 70 (2012),435- 448.

An outline of the paper is as follows: Chapter 2 contains a very basic and friendly introduction to graphs and groups, though focusing particular attention on Cayley graphs, permutation groups, automorphism groups and wreath products. We begin chapter 3 with some nice examples of isomorphic Cayley graphs on non-isomorphic groups. We continue with some interesting results from the literature, and end the chapter as well as this thesis with the proof of our main theorem. Chapter 2

Preliminaries

In this chapter we will review the deﬁnitions and results from graph and group theory respectively that will be needed in the thesis. For a deeper look into graph theory, see [2]. For more information about permutation groups, see [4] and for more about algebra, I would recommend [7] and [8]. For those desiring a more intuitive approach to learning group theory, I highly recommend [3].

2.1 Graphs

2.1.1 Basic deﬁnitions

A simple graph, also called an undirected graph X = (V,E) consists of a vertex set V = V (X) and an edge set E = E(X), where each edge is an unordered pair

{v1, v2} of distinct vertices, v1, v2 ∈ V . The graphs in Figure 2.1 are all simple and undirected graphs. Two vertices which are contained in an edge are said to be adjacent to one another and incident to that edge. Two edges with a vertex in common are also said to be incident to one another. If X = (V,E) and X0 = (V 0,E0) are graphs and V 0 ⊆ V and E0 ⊆ E, then X0 is said to be a subgraph of X, and we write X0 ⊆ X. If X0 ⊆ X then any pair of vertices of V 0 which are adjacent in X0 are adjacent as well in X. If, moreover, any

0 0 pair v1 and v2 of vertices in V which are adjacent in X are also adjacent in X , then

4 Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 5 the subgraph X0 is said to be full or induced. The degree of a vertex is the number of edges to which it is incident. The sum of the degrees over all vertices in a graph is equal to twice the number of edges in the graph. A graph is called r-regular if every vertex has degree r. Two very important graphs are cycles and paths, mostly because of their roles as subgraphs. The path Pn on n vertices is the graph with V (Pn) = {1, . . . , n} and

E(Pn) = {{1, 2}, {2, 3},..., {n − 1, n}}. The vertices 1 and n are the only vertices of degree 1 in the path, and are said to be end vertices, see Figure 2.1. Similarly, the cycle Cn on n vertices, n ≥ 3, is a the graph with V (Cn) = {1, . . . , n} and

E(Cn) = {{1, 2}, {2, 3},..., {n − 1, n}, {n, 1}}. Cycles are 2-regular graphs, in fact, any 2-regular graph is either a cycle or a disjoint union of cycles.

Figure 2.1: Some simple graphs.

A graph X is connected if each pair of vertices are the end vertices of some subgraph which is a path. We say simply that the vertices are joined by a path. More generally, digraph X consists of a vertex set V (X) and a set E(X) of arcs (sometimes denoted A(X)), or ordered pairs of distinct vertices. In general, an arc (x, y) is represented in ﬁgures by an arrow from x to y. If both (x, y) and the opposite pointing arc (y, x) are in E(X), the two arcs can be represented by a single Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 6 undirected edge. Even more generally, in a multigraph the edge set consists of ordered pairs of vertices, such that each arc is given a multiplicity. Thus, both loops, or edges of the form (x, x), and multiple edges between pairs of vertices are permitted. Two (di)graphs X and Y are said to be isomorphic if there exists a bijection φ : V (X) → V (Y ) between the vertex sets such (x, y) ∈ E(X) if and only if (φ(x), φ(y)) ∈ E(Y ). For isomorphic (di)graphs X and Y we use X ∼= Y . In particular, an automorphism is an isomorphism of a graph onto itself. The automorphisms of a graph form a group, the automorphism group discussed in the next section. In this thesis, we will only consider simple graphs and digraphs, so multiple edges or loops will not be allowed, however disconnected graphs are allowed.

2.2 Groups

2.2.1 Basic deﬁnitions

A group is a set G together with a binary operation ? : G × G → G, written

?(g1, g2) = g1 ? g2, which is associative, g1 ? (g2 ? g3) = (g1 ? g2) ? g3, is such that there is an identity element e ∈ G with e ? g = g ? e = g for all g ∈ G, and is such that each element g ∈ G has an inverse element g0 ∈ G such that g ? g0 = g0 ? g = e. It is common practice that if the multiplication is also commutative, g1 ? g2 = g2 ? g1 for all g1, g2 ∈ G, then the operation is termed an addition, written with ‘+’, the identity element is denoted by ‘0’, the inverse element of g is denoted by −g, and the group is said to be abelian. Otherwise, in the case of a not-necessarily abelian group, the operation is regarded as multiplication, the identity is denoted by ‘1’, and the inverse of g is written as g−1.

A group homomorphism is a function f : G1 → G2 between the underlying sets of two groups G1 and G2 which respects multiplication, inverses, and the identity, i.e. f(g ? h) = f(g) ? f(h); f(g−1) = f(g)−1; f(1) = f(1).

If the function is bijective, the mapping is an isomorphism. In general, the set of Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 7

elements Kf = {g ∈ G | f(g) = 1}, the kernel of f is a normal subgroup of G. For ∼ isomorphic groups G1 and G2 we write G1 = G2. A group G is said to be generated by a subset S ⊆ G if every element g ∈ G can be expressed in terms of the elements of S, their inverses, and the group multiplication. The simplest examples are the cyclic groups which are generated by a single element. Cyclic groups are necessarily abelian, so were let 1 denote the generator, then there are two cases. If there are an inﬁnite number of elements the group is isomorphic to the additive group Z of integers. If there are only ﬁnitely many elements then the cyclic group is isomorphic to Zn, the additive group of integers modulo n. The cardinality of the set G is called the order of the group.

A (proper) subset G1 ⊆ G of a group G which is closed under the operation and the taking of inverses is said to be a (proper) subgroup of G, and G is said to be an overgroup of G1, and we write G1 ≤ G. For example, the cyclic group Z15 has four ∼ ∼ subgroups, the trivial group {0}, 5Z15 = {0, 5, 10} = Z3, 3Z15 = {0, 3, 6, 9, 12} = Z5, and Z15 itself. Every element g of a group G generates a cyclic subgroup of G, and order of that cyclic subgroup is called the order of the element g. Given a group G, a subgroup H ≤ G, and an element g ∈ G, the set gH = {gh | g ∈ H} is a left coset of H. Clearly |gH| = |H| and it is easy to show that the left cosets of G by H partition the set G, establishing Lagrange’s Theorem, that the order of a subgroup divides the order of a group. In particular, the order of any element of G divides the order of G. Similarly, one can deﬁne right cosets Hg = {hg | h ∈ H}, and if gH = Hg for all g ∈ G, then the subgroup is said to be normal. If H is a normal subgroup then

(gH)(g0H) = g(Hg0)H = (gg0)(HH) = (gg0)H so the multiplication of cosets is well-deﬁned giving the set of cosets, G/H a group structure induced from the multiplication of G. The group of cosets G/H is called a factor group.

Given two groups G1 and G2, their cartesian product is the group G1 ×G2 whose elements are the set of ordered pairs

{(g1, g2) | g1 ∈ G1, g2 ∈ G2} Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 8 with multiplication deﬁned in each coordinate separately,

0 0 0 0 (g1, g2)(g1, g2) = (g1g1, g2g2).

Clearly if G1 and G2 are both abelian, then G1 ×G2 is abelian. For example, Z3 ×Z5 is abelian and has ﬁfteen elements and is generated by the set {(1, 0), (0, 1)}. It is, in fact, cyclic, generated by {(1, 1)}, and

∼ Z15 = Z3 × Z5.

In general, given two primes p and q,

∼ Zpq = Zp × Zq.

∼ On the other hand Z9 6= Z3 × Z3, since Z3 × Z3 only has elements of order 1 and 3. In general we have the following.

Theorem 2.2.1 (Fundamental Theorem of Abelian Groups). Every ﬁnitely generated abelian group G is isomorphic to a product of cyclic groups of inﬁnite or prime power order ∼ ∼ n G = × · · · × × 1 × k = × 1 × k Z Z Zp1 Zpk Z Zp1 Zpk where the i are positive integers and the pi are not-necessarily distinct primes.

Moreover, the expression is unique provided that pi ≤ pi+1 and i ≤ i+1 for those exponents associated with the same prime.

An elementary abelian p-group is a ﬁnitely generated abelian group all of whose elements are of order p, so only one prime p occurs in its expression (see Theorem 2.2.1). Of particular interest in this thesis are the 2-groups, all of whose elements are of 2-power order.

2.2.2 Automorphism groups

An important source of groups is the set of automorphisms of a structure. Specif- ically, given a graph X = (V,E), the set of all automorphisms of X clearly forms a group under composition. It is significant that the converse of this observation Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 9 is also true, as expressed in Frucht’s Theorem which says that every group is the automorphism group of some graph. It is illustrative to attempt to find a graph X such that Aut(X) ∼= G for some G. For the trivial group, we can take, for instance, the one vertex graph, otherwise known as the trivial graph, but any graph with no non-trivial automorphisms is sufficient. A stronger version of Frucht’s theorem proved a decade later states that any group is the automorphism group of a cubic graph, and in fact the smallest cubic graph with a trivial automorphism group was named after Frucht. The Frucht graph is shown in Figure 2.2

Figure 2.2: The Frucht graph

For a slightly more substantial example, let us take a look at the cyclic groups. Because of the simple structure of such groups, it is easy to ﬁnd graphs whose automorphism groups have a subgroup isomorphic to a cyclic group. In particular, any graph exhibiting rotational symmetry has an automorphism group with such a subgroup.

For example, an automorphism of the cycle graph Cn

0 1 n-1 2

n-2 3

n-3 4 ... 5 is determined by the images of the vertices 0 and 1. There are n possible images of

0, and each is realized by one of the “rotations” ρk(i) = i+k, and setting τ(i) = −i, Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 10

the n “reflections” ρkτ complete the 2n automorphisms found the automorphism group of the n-cycle graph, defining the dihedral group D2n. The dihedral group for n > 2 is the simplest example of a non-abelian group, and the subgroup of the dihedral group induced by rotations is always a cyclic group. In order to find a graph whose automorphism group is isomorphic to a cyclic group Zn with n > 2 we have to find a way to allow rotations while preventing reflections or other unwanted automorphisms. To do this we need more vertices, at least twice as many in fact. The smallest graph whose automorphism group is Z3 is shown in Figure 2.3 and had 9 vertices.

Figure 2.3: The smallest cyclic group graph

One way to easily find (not necessarily the smallest) graphs with the automorphism group Zn is described by Lovàszin his selection of combinatorial problems and exercises [15]. Construct a cycle of length 3n and append paths of length 1 and 2 to every third vertex. Because any reflection can not map paths of the same length onto each other, rotations are the only automorphisms, thus yielding the desired automorphism group. The first three such graphs are shown in Figure 2.4.

Figure 2.4: Graphs with the automorphism groups Z3, Z4, and Z5.

To end this section, we will construct connected graphs whose automorphism Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 11

n ∼ groups are the elementary abelian groups Z2 . It is easy to check that Aut(K2) = Z2. 2 For Z2 we might try to take two copies of K2, but this graph has the ”unwanted” automorphism of switching the copies, and has an isomorphism group isomorphic to the dihedral group of order 4. This is because the complement of two copies of K2 is precisely C4, and since an automorphism maps edges to edges and nonedges to nonedges, Aut(G) ∼= Aut(G). To prevent the appearance of such unwanted automorphisms, we need to ﬁnd non-isomorphic graphs each with the same automorphism

n group, and a simple class of such graphs are the paths. Thus Z2 is the automorphism group of a (disconnected) graph consisting of n paths of diﬀerent lengths. The complement of a disconnected graph is connected, and the ﬁrst three graphs constructed in this way are shown in Figure 2.5.

2 3 Figure 2.5: The automorphism groups of these graphs are Z2, Z2 and Z2 respectively.

2.2.3 Permutation groups and group actions

Let S be a finite set. Then Sym(S) denotes the set of all bijections, or permutations of S. The set Sym(S) is a group under composition, with cardinality | Sym(S)| = |S|!. Any subgroup of Sym(S) is also called a permutation group, so in particular the automorphism group of a graph is a permutation group and Cayley’s Theorem implies that every finite group is isomorphic to a permutation group. More generally any group homomorphism φ : G −→ Sym(S) establishes a cor- respondence between the elements of G and a set of permutations of S, and we say that G acts on S. Often the notation for the homomorphism φ is suppressed and we write φ(g)(s) = gs. If the homomorphism φ is one-to-one, so only the identity element of g acts as the identity on S, the action is said to be faithful. The action is Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 12 transitive if for any pair of elements x, y ∈ S there exists a g ∈ G such that gx = y. An action is free if the only element of G which fixes any element of s is the identity, or, equivalently, if gs = hs for some s ∈ S implies g = h. If the group G acts on S, then given any element s ∈ S, the orbit of s under G

G is the set s = {gs | g ∈ G} and the stabilizer of s is the set Gs = {g ∈ G | gs = s. For each element s ∈ S it is easy to see that

G |G| = |s | · |Gs|.

One important example of a group action is the action of a group G on the set of (left) cosets G/H of G with respect to some subgroup H, defined by g(xH) = (gx)H. If H is the trivial subgroup consisting only the identity element, this action is called the left regular action. As another example, consider the group of symmetries of the cube. The group acts transitively but not freely on the set of vertices, since, for instance, there is 120◦ rotation about any pair of antipodal vertices which fixes both of them, but no other vertices. This group acts transitively as well on the set edges, and the set of faces. There is a transitive action as well on the set of arcs, where an arc is a pair (v, e) consisting of one vertex, v and one edge, e which is incident to v. This action of the symmetry group on the arcs is also not free, since the mirror of a diagonal reflection across a plane contains four arcs, each of which are fixed, while all other arcs are not fixed. The symmetries of the cube also acts on the set of flags, that is, the set of triples (v, e, f) where v is a vertex, e and edge incident to v, and f is a face containing e, which can be visualized as a triangular section of each face, see Figure 2.6. This action is both transitive and free. Note that the group of rotational symmetries of the cube acts freely but not transitively on the set of flags, since there is no rotation which will map a white triangle onto a green triangle. An action which is both free and transitive is called regular. A regular action makes it possible to identify in a very natural way the objects of the set S with the elements of the group G. Specifically, arbitrarily choose one element x of S and label it with the identity element 1 of G. Then label each of the other elements of S with the unique element g of G which maps the element labeled 1 to it, so if g(x) = y, Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 13

Figure 2.6: The symmetries of the cube act on the vertices, the edges, the faces, the arcs, and the flags of the cube. label y with g. Now g0(y) = g0(g(x)) = (g0g)x, so the action of any element g0 is to take the element labeled g to the element labeled g0g. Thus the labeling, in other words the choice of x, identifies the given action with the left regular action defined above.

2.2.4 Imprimitivity block systems

Let G be a group acting transitively on a set Ω, and let B be an imprimitivity block system (called block system for short) of G. Denote by GB the corresponding kernel, that is, GB = {g ∈ G | g(B) = B for all B ∈ B}. Furthermore, denote the permutation group of B induced by the action of G on B by GB. For a block B, denote by G{B} the set-wise stabilizer of B in G, and for a subgroup H ≤ G{B}, denote the restriction of H to B by HB.

For an arbitrary group G and g ∈ G, let gL be the permutation in Sym(G) deﬁned by g(x) = gx, x ∈ G. Thus GL = {gL | g ∈ G} is the left regular representation of

G. Throughout the thesis a permutation group X ≤ Sym(G) with GL ≤ X will be referred to as an overgroup of GL in Sym(G). We shall use the following properties of blocks (see [13, Lemma 3.1]).

Proposition 2.2.1. Let A be a transitive permutation group on a set Ω which contains a regular abelian subgroup G, and let B be a block system of A. Then for ∼ every block B ∈ B, G∩AB is faithful and regular on B, and GAB/AB = G/(G∩AB) is regular on B.

The following fact follows easily from the above proposition. Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 14

Corollary 2.2.2. Let A be an overgroup of GL in Sym(G), where G is an abelian group. Then every block of A must be an H-coset for some subgroup H ≤ G.

We shall make use of the following simple observation.

Proposition 2.2.3. Let A be an overgroup of GL in Sym(G) and T be an orbit of the point stabilizer Ae of the identity e of G in A. Then the subgroup {g ∈ G | gT = T } ≤ G is a block of A.

Proof. Set H = {g ∈ G | gT = T }. Consider the set-wise stabilizer A{T } of T in A. For a ∈ A we can write a = gLa1, where gL ∈ GL and a1 ∈ Ae. Thus a is in A{T } exactly when T = a(T ) = (gLa1)T = gL(a1(T )) = gL(T ) = gT . Therefore

HL Ae A{T } = HL Ae. Since Ae ≤ HL Ae, the orbit e is a block of A (see [4, Theorem

1.5A]). As eHL Ae = eAe HL = H, the proposition follows.

For a subset S ⊆ G and integer i we let S(i) = {si | s ∈ S}. The following result is a special case of [18, part (a) of Theorem 23.9].

Theorem 2.2.4. Let A be an overgroup of GL in Sym(G), where G is an abelian group, and let T be an orbit of the point stabilizer Ae of the identity e of G in A. (i) Then for every integer i with (i, |G|) = 1, T is also an orbit of Ae.

2.2.5 Wreath products of groups

Suppose we have a connected graph X = (V,E) with automorphism group G. Con- S sider the disconnected graph i Xi consisting of n components Xi = (Vi,Ei) each iso- S morphic to X. Then each automorphism of i Xi permutes the components. Those automorphisms which ﬁx the components are determined by n automorphisms, automorphism gi ∈ G in the i’th component. In general such an automorphism will be followed by a permutation π of the components. So letting (v, i) denote a vertex in the i’th component, the automorphism with data (π; g1, . . . , gn) will act via

(π; g1, . . . , gn)(v, i) = (gi(v), π(i)) Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 15 and the composition rule will be given by

0 0 0 0 0 0 (π ; g1, . . . , gn)(π; g1, . . . , gn) = (π π; gπ(1)g1, . . . , gπ(n)gn)

This calculation motivates the more general wreath product of groups, in which, instead of allowing the full permutation group Sym(n) of the indices of the components, we restrict to a subgroup, or, equivalently, a group with a faithful action on {1, . . . , n}. Formally, the wreath product of a permutation group Π ⊆ Sym(n) and a group G is denoted by Π o G and consists of the |Π| · |G|n elements of the form

(π; g1, . . . , gn) and multiplication deﬁned by the composition rule above. An important special case is that the permutation group π arises from a group acting on itself by the left regular action. It is not hard to modify the disjoint union of graphs to obtain a graph whose automorphism group realizes the wreath product. The most direct construction is the wreath product of graphs. Given two graphs (digraphs) X1 and X2, the wreath product X1 o X2 is the graph (digraph) with vertex set V (X1) × V (X2), and an edge

(arc) from (x1, x2) to (y1, y2) if and only if

• (x1, y1) is an arc of X1, or

• x1 = y1 and (x2, y2) is an arc of X2.

For example, in Figure 2.7 we ﬁnd 8 copies of the tetrahedron graph arranged

Figure 2.7: The wreath product of the graph of the tetrahedron, T , with the graph of the cube, C, giving C o T . in a wreath product. As observed above, the individual tetrahedron graphs can be independently acted on by their automorphism groups, however, they can now only Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 16 be permuted by a permutation which respects the incidences of the other factor, the cube graph. Clearly, the automorphism group of C o T is Aut(C) o Aut(T ).

In general, the wreath product Aut(X1) o Aut(X2) of the groups Aut(X1) and

Aut(X2) acts as a group of automorphisms of the wreath product X1 o X2 of the graphs X1 and X2. It may happen, say in the cases of complete graphs, that

Aut(X1) o Aut(X2) is only a proper subgroup of the full automorphism group. Here we follow the convention that Aut(X1) o Aut(X2) has active group Aut(X1) and passive group Aut(X2), so the order of Aut(X1) o Aut(X2) satisﬁes

|X1| | Aut(X1) o Aut(X2)| = | Aut(X1)| · | Aut(X2)| .

The asymmetry of the wreath product, that is the fact that it is not necessarily true that A o B ∼= B o A, is illustrated in Figure 2.8 which depicts T o C.

Figure 2.8: The wreath product T o C.

A less cluttered construction we call the reduced wreath product, X1 or X2. It has vertex set V (X1) ∪ (V (X1) × V (X2)) and arcs

0 0 • (x1, x1) if (x1, x1) is an arc of X1,

• ((x1, x2), (x1, y2)) if (x2, y2) is an arc of X2, and

• ((x1, x2), x1) for all x1 ∈ V (X1) and x2 ∈ V (X2).

At the cost of |V (X1)| more vertices the reduced wreath product realizes the wreath product of the automorphism groups with only

|E(X1)| + |V (X1)||E(X2)| + |V (X1)||V (X2)| Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 17 edges instead of

2 |V (X1)||E(X2)| + |E(X1)||V (X2)| for the ordinary wreath product of graphs. The reduced wreath products of the examples C and T as well as T and C are illustrated in Figures 2.9 and 2.10 respectively.

Figure 2.9: The reduced wreath product C or T .

Figure 2.10: The reduced wreath product T or C.

2.3 Cayley graphs

Given a group G and a generating set S, the Cayley graph on G with respect to S is the group with vertex set G and arc set {(g, gs) | g ∈ G, s ∈ S}. The Cayley graph is a directed graph, if S 6= S−1. If the set S is a generates the group, the Cayley graph is connected, otherwise it is disconnected, and its components are isomorphic to Cay(hSi,S). Sometimes the generating set S is called the connection set. The absolute simplest Cayley graphs to describe and to analyze are those corresponding to cyclic groups, see Figure 2.11 for some Cayley graphs of Z12. Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 18

0 1 0 1 0 1 11 2 11 2 11 2

10 3 10 3 10 3

9 4 9 4 9 4 8 5 8 5 8 5 7 6 7 6 7 6

Figure 2.11: Cayley graphs for Z12 with connection sets consisting of {1}, {1, 11}, and {3, 4}.

An important feature of Cayley graphs is that the left regular action on group preserves the arcs, so the group not only acts as a group of automorphisms of the Cayley graph, but acts freely and transitively on the set of vertices. The converse of this observation is also true, that if a group acts as a group of automorphisms on a connected graph such that the action is free and transitive on the vertex set, that graph is a Cayley graph for that group. Free and transitive actions are a fruitful source of Cayley graphs, for example the free and transitive action on the flags of the cube of the example in Figure 2.6 can be used to create a Cayley graph of the symmetry group of the cube, with one vertex for each flag, and the connection set consisting of reflections across the boundary lines of a fixed triangle.

Figure 2.12: A Cayley graph constructed from a free transitive action of the symmetry group of the cube on the set of triangular regions subdividing each of the six faces. Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 19

2.3.1 Arc transitive Cayley digraphs on Cyclic groups

Let X be a Cayley digraph Cay(G, S) on a group G. In case S = S−1 we shall regard X also as an undirected graph. It is immediate to see that the left regular representation GL is a group of automorphisms of X acting regularly on the vertices.

Aut(X)e Denote by Aut(X)e the stabilizer of the identity e of G in Aut(X), and let x denote the orbit of x under Aut(X). The orbital digraphs of Aut(X) are the Cayley digraphs Cay(G, x Aut(X)e ), x ∈ G. Clearly, all orbital digraphs of Aut(X) are arc- transitive. Now suppose that G is an abelian group, let H be a subgroup H ≤ G. The quotient digraph X/H is the simple digraph where the vertex set is the factor group

G/H, and (Hx, Hy) is an arc if and only if X has an arc (h1x, h2y) for some h1, h2 ∈ H. Note that, we do not allow X/H to have loops and multiple edges. Note also that X/H is a Cayley digraph on the group G/H, and it is given as

X/H ∼= Cay(G/H, S/H), where S/H = {Hs | s ∈ S}.

The following classiﬁcation of arc-transitive Cayley digraphs on Z2n can be de- duced from [11, Theorem 4] (for a classiﬁcation of all arc-transitive Cayley digraphs on cyclic groups, see also the papers [14, 17]).

Theorem 2.3.1. Let X = Cay(Z2n ,S) be a connected Cayley digraph. Then X is arc-transitive if and only if S is one of the following sets:

−1 2n−1−1 Z2n \ H, Hz, Hz ∪ Hz , or Hz ∪ Hz with n ≥ 2, where z is a generator of Z2n and H is a subgroup of Z2n , 1 ≤ H < Z2n .

2.3.2 CI-Graphs

The problem of when a Cayley graph can have a diﬀerent representation over the same group, that is Cay(G, S) = Cay(G, T ), is called the Cayley Representation problem.

It is easy to see that the cycle Cn can be realized as a Cayley graph over the group Zn. However, there are a number of diﬀerent Cayley representations over Zn Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 20 of this graph. That is, the diﬀerent choices for the generating set are determined − ∼ by n and the prime factors of n. In fact, Cay(Zn, {k, k 1}) = Cn whenever hki =

Zn. This motivates the notion of equivalent Cayley representations. In fact, two Cayley representations are equivalent if an automorphism of the vertex set maps one generating set onto the other. By coloring the edges consistently with respect to their generators, this is easy to visualize. Equivalent representations of Cayley graphs are those which can be colored the same way. For example, the graph Cay(Z8, {1, 7, 4}) has two generators of order 8 and one of order 2, and Z8 has only one other candidate for a generator of order 8, but no other choice for a generator of order 2, so there are only two Cayley representations of this graph, and they are equivalent. One might be tempted to assume that any two representations of a circulant are equivalent. This was the subject of a conjecture of Ad´am[1]´ in the sixties disproved by Elspas and Turner [6] some years later. Take, for instance, the circulant

Cay(Z16, {1, 2, 7, 9, 14, 15}) and Cay(Z16, {2, 3, 5, 11, 13, 14}). The generating sets are not equivalent, and as seen in the ﬁgure below, the isomorphism maps green onto both red an blue, so the representations are not equivalent. This (disproved) conjecture is what spurred the research of so called Cayley Isomorphism graphs or CI-graphs, that is, Cayley graphs whose representations are all equivalent.

5 2 0 7 3 4 14 9 1 6 12 11 15 8 10 13

Figure 2.13: This isomorphism shows that Cay(Z16, {1, 2, 7, 9, 14, 15}) and

Cay(Z16, {2, 3, 5, 11, 13, 14}) are inequivalent representations of a circulant.

CI-graphs are of interest primarily because they are useful in discovering isomorphism classes of Cayley graphs. To determine whether two graphs are isomorphic is Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 21 of course harder than to determine whether two generating sets are equivalent, so such classes are useful in the study of Cayley graphs. Chapter 3

Non-isomorphic groups with isomorphic Cayley graphs.

Given a finitely generated group G there will be a Cayley graph associated with every finite generating set S of G. In general, a given group will have many non-isomorphic Cayley graphs, even for generating sets with same cardinality. By contrast, given a graph, we can ask what conditions ensure, in the first place, that the graph is the Cayley graph of a group, and, in the second place, that that realization is unique. It is this second question which concerns us in this thesis, and in this chapter we will give a brief overview of what is known. The answer to the existence part of the question has been answered classically by Sabidussi:

Theorem (Sabidussi). A graph X is a Cayley graph if and only if a subgroup G of its automorphism group acts freely and transitively on the vertex set.

If the free transitive subgroup is the whole automorphism group of the graph, then that graph is uniquely describable as a Cayley graph. On the other hand, if a free transitive subgroup G 6= Aut(X), so necessarily the vertex stabilizers in Aut(X) are non-trivial, it may or may not be possible to ﬁnd a non-isomorphic subgroup of Aut(X) which also acts freely and transitively on V (X). In the next section we will give some examples of graphs which can be realized as Cayley Graphs on at least two (non-isomorphic) groups.

22 Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 23

3.1 Inﬁnite graphs and free groups

An interesting case is that of a free group. If F is a free group freely generated by a set S, the Cayley graph of F with respect to S is an infinite 2|S| valent tree. The vertex stabilizers in the automorphism group of a 2|S| valent tree are infinite, so it might be supposed that there are many opportunities for the existence of a different free transitive subgroup, however a theorem of Serre states:

Theorem 3.1.1. Every group which acts freely on a tree is free.

Note that to “act freely on the tree” the group must act freely on both the vertices and the edges. If you relax that condition on the edges, so one allows edges to be ﬁxed, then, since the action is still free on the vertices, the endpoints of a ﬁxed edge must be interchanged under an automorphism. Such an action on a tree may be free, and in fact transitive on the vertices, yielding a realization of the tree as a Cayley graph of a non-free group, see Figure 3.1. Note that the upper graph is

b a c

Figure 3.1: The lower digraph is part of an infinite 2-valent tree, freely acted on by the free rank 1 infinite cycle group of translations generated by c. The upper graph is acted on by a group isomorphic to the free product Z2 ?Z2 generated by the involutory ‘reflections’ a and b. drawn as a simple graph and the lower graph is drawn as a digraph, so to establish the isomorphism we would either disregard the directions in the digraph or regard the undirected edges in the upper graph as matched pairs of directed edges, and add c−1 to the generating set for the lower Cayley graph. In general we will not point out such obvious modifications. Another treelike example is illustrated in Figure 3.2 where the Cayley graphs are isomorphic, with the isomorphism preserving the directions on the edges, and yet the groups can be shown to be non-isomorphic. The group on the left has elements of order three, and is in fact clearly the free product Z3 ? Z3 ? Z3. The group on the Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 24

Figure 3.2: The simple digraph on the left depicts a small section of the infinite Cay- ley graph associated to the group with presentation ha, b, c | a3 = b3 = c3 = 1i. The recolored digraph the right depicts the Cayley graph of the group with presentation ha, b, c | abc = 1i. right has all non-trivial elements of infinite order. In fact, the group on the right is a free group, freely generated by, say, a and b, with the equation abc = 1 merely identifying the third generator c redundantly as the element c = b−1a−1. In terms of the graph, if on deletes the red edges corresponding to the redundant generator c, the result is an infinite four valent tree. The previous examples suggest that examples may be found by considering graphs which are not only vertex transitive, but edge transitive as well, for instance, the graphs of plane lattices and platonic solids. The graph of the integer square lattice with edges oriented counterclockwise around every square face has two distinct actions which are free and transitive on the vertices, see Figure 3.3. The group of the action on the left is generated by two 90◦ rotations a and b. The group of the Cayley graph on the right is generated by two perpendicular glide reflections of the lattice. So as transformation groups they are not the same, however are they non-isomorphic? The technique of the previous example is not sufficient since these two groups have the same abelianizations. In the abelianization of ha, b | (ab−1)2 = (ab)2 = 1i both a and b are of order 4, since 1 = (ab−1)2 · (ab)2 = a4 and 1 = (ab−1)−2 · (ab)2 = b4, while in the abelianization ha, b | a4 = b4 = (ab−1)2 = 1i we have (ab)2 = (ab)2 · b−4 = (ab−1)2 = 1. So both have abelianization

−1 Z4 × Z2 = hai × hab i. Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 25

z y

b b a a 1 x

Figure 3.3: Two Cayley graphs on the graph of the integer plane lattice. The group on the left is presented by ha, b | a4 = b4 = (ab−1)2 = 1i, and the group on the right is presented by ha, b | (ab−1)2 = (ab)2 = 1i.

Nevertheless, we can use the action of the group on the lattice to show that ha, b | (ab−1)2 = (ab)2 = 1i has no element of order four, even though its abelianization does have such elements. Every element in ha, b | (ab−1)2 = (ab)2 = 1i corresponds to a path from 1 along the colored integer lattice ending in a vertex whose directed edges in the lattice are in one of four possible configurations, indicated by 1, x, y or z in Figure 3.3, depending on whether the green arrows are oriented to the left or the right, and whether the red arrows are oriented up or down. If the path from 1 ends at another vertex (n, m) of the same type as 1, then repeating that path will result in another vertex (2n, 2m), also of type 1, and so on, with that element of infinite order. If the path from 1 ends at a vertex of type y, so the orientations are exactly reversed, then repeating that vertex will return to 1, and the element is of order 2. If the path from 1 ends at a vertex (n, m) of type x, with the same red orientation and reversed green orientation, then repeating that path starting at x will result in (2n, 0), continuing to (3n, m), etc, so again we have an element of infinite order. In the same way, a the path from 1 which ends at a vertex of type y corresponds to a group element of infinite order. So all elements of ha, b | (ab−1)2 = (ab)2 = 1i have order 1 or 2 or ∞. Geometrically, the elements act on the lattice by glide reflections (types x and z), half-turns (type z) and translations (type 1). The triangular lattice with all triangles oriented counterclockwise can also be Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 26 colored in two ways to form Cayley graphs, see Figure 3.4. Here it is fairly easy

Figure 3.4: Two Cayley graphs on the graph of the triangular plane lattice. On the left the group has presentation ha, b, c | a3 = b3 = c3 = abc = 1i while on the right the group has presentation ha, b, c | abc = bca = 1i to show that the underlying groups are non-isomorphic. In the group on the left the generator a is redundant, the equation abc = 1 identiﬁes a with (bc)−1 and the remaining equation cba = 1 states 1 = cb(bc)−1 = cbc−1b−1, in other words, the only relation between b and c is that they commute, so

ha, b, c | abc = bca = 1i = hb, c | bc = cb = 1i = hbi × hci = Z × Z a free abelian group of rank 2, having no elements of order 3. The graph of the regular hexagonal lattice has two colorings as Cayley graphs in which all the undirected edges refer to involutions, see Figure 3.5. The groups may be shown to be non-isomorphic, again by considering their abelianizations. In the abelianization of ha, b, c | a2 = b2 = c2 = (ab)3 = (bc)3 = (ca)3 = 1i the relation 1 = (ab)3 = ababab = a3b3 = ab says simply that generators a and b are equal. The three relations (ab)3 = (bc)3 = (ca)3 = 1 therefore imply that a = b = c, and the group is

2 2 2 2 isomorphic to Z2. For the abelianization of ha, b, c | a = b = c = (abc) = 1i the relation (abc)2 = 1 is a consequence of the other three, so the group is isomorphic to Z2 × Z2 × Z2. There is another coloring of graph of the hexagonal lattice in which only one of the generators is an involution, see Figure 3.6. Writing c = ab, this is equivalent to ha, c | a2 = (ac)6 = c3 = 1i, with the relation (ac)6 = 1 a consequence of the other Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 27

b a b a

c c

Figure 3.5: Two Cayley graphs on the graph of the regular hexagonal lattice. The group on the left is presented by ha, b, c | a2 = b2 = c2 = (ab)3 = (bc)3 = (ca)3 = 1i and the group on the right is presented by ha, b, c | a2 = b2 = c2 = (abc)2 = 1i.

b a

Figure 3.6: Another Cayley graphs on the graph of the regular hexagonal lattice. The group is presented by ha, b | a2 = b6 = (ab)3 = 1i. two in the abelianization, so the group abelianizes to Z2 × Z3, and the graph of the hexagonal lattice is the Cayley graph of three non-isomorphic groups.

3.2 Graphs of the Platonic solids

There are ﬁve Platonic solid graphs, corresponding to the tetrahedron, the cube, the octahedron, the icosahedron, and the dodecahedron. Four of them realize Cayley graphs of non-isomorphic groups. For the graph of the tetrahedron, see Figure 3.7, the vertices each have valence 3, so for the graph to be a Cayley graph, the generators for the group would have to either be three involutions, partitioning the edges into three perfect matchings, which decomposition corresponds to the Cayley graph Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 28

Figure 3.7: Two realizations of the tetrahedral graph as a Cayley graph. On the left the Klein four group acts freely and transitively on the vertex set by three 180◦ rotations along the three coordinate axis. On the right a 90◦ rotary reflection about an axis through the centroids of one pair of antipodal edges generates a cyclic group isomorphic to Z4 acting freely and transitively on the vertices. The second generator is the square of the first. on the left of the figure; or the generators consist of one involution and one non- involution. Deleting a perfect matching of two edges from the tetrahedron graph yields a simple 4-cycle, so the only other possibility is an involution and an element of order four, as see on the right of the Figure. We may similarly analyze the possible realizations of the graph of the cube as a Cayley graph. Like the tetrahedron, each vertex has valence 3, so there could be either one or three involutions. If the involution corresponds to a parallel class of edges, say the red edges of Figure 3.8, The eight remaining edges are partitioned into two 4-cycles. If there are two oriented 4-cycles, necessarily belonging to the same generator, either the cycles have the same or contrasting orientations, illustrated in the left two Cayley graphs of Figure 3.8, in which case the group will be either isomorphic to the direct product Z4 × Z2 or the dihedral group D8. The edges of the two 4-cycles may also correspond to involutions, such as the Cayley graph on

3 the right of Figure 3.8, giving a group isomorphic to Z2. There is yet another possibility for the cube. There is a second type of perfect matching in the graph of the cube, illustrated by the red edges of Figure 3.9. In this case the involution corresponding the red edge cannot be a rotation, since that would not preserve the set of red edges, and that transformation must be a Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 29

Figure 3.8: Three realizations of the graph of the cube as a Cayley graph. On the left, a 90◦ face rotation together with 180◦ edge rotation whose axis lies perpendicular to the first axis, generate a subgroup isomorphic to D8, having eight elements acting freely and transitively on the vertices. In the middle, a 90◦ face rotation together with a reflection in a perpendicular plane generate a subgroup of order 8 isomorphic to Z4 ×Z2. On the right reflections in three perpendicular planes generate a subgroup of symmetries isomorphic to Z2 × Z2 × Z2. reflection, in which case one of the remaining generators, and hence both, must be involutions, and, in fact, the graph must be colored as in Figure 3.9. The red and blue reflection a and b generate a copy of the Klein 4-group and the green generator is a rotation, and the group of this Cayley graph has the presentation ha, b, c | a2 = b2 = c2 = (ab)2 = acbc = 1i. The relation acbc = 1 can be written either c−1bc = a or c−1ac = b, which identifies the group with the semi-direct product hci n (hai × hbi) which in this particular case also identifies the group as the wreath product hci o hai. Notice that the group ha, b, c | a2 = b2 = c2 = (ab)2 = acbc = 1i is, in fact, isomorphic to the dihedral group D8 of order 8. So the graph of the cube is a Cayley graph over three non-isomorphic groups, and we have found seven representations so far, each corresponding to the choice of involutions in the first step. There is however one more case which contains again 3 more representations. In the Cayley graph on the right of Figure 3.8 the generators have been taken to be three reflections. Clearly, if instead of reflections we took 180◦ edge rotations, these involutions would also preserve each of the three colors, and ought to be considered as a candidate for a Cayley graph. However, these three transformation Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 30

Figure 3.9: A fourth realization of the graph of the cube as a Cayley graph. Here two of the involutory perfect matchings are not all parallel classes of edges in the cube. The red and blue edges correspond to reflections a and b in the yellow planes, and together generate a subgroup isomorphic to hai × hbi = Z2 × Z2, and the green generator corresponds to a 180◦ rotation which flips the reflective planes. do not generate a group which acts freely on the vertices of the cube. If fact, the product of any two of them is a 120◦ rotation fixing a pair of vertices of the cube. So the action on the graph of the cube does not realize that graph as a Cayley graph. To see how this transitive but not free vertex action is distinct from the actions which correspond to Cayley graphs, consider the action on the semi-edges. This action is free, and if we connect the 24 semi-edges with colored edges corresponding to the three rotation generators, see Figure 3.10, we obtain a quite natural but

I I

Figure 3.10: Three edge rotations each separately generate a free action on the vertices, but collectively generate the full rotational group of the cube, as is seen by taking the vertices at the endpoints of the middle thirds of the edges of the cube. rather unpopular drawing of the Cayley graph of the rotational group of the graph of the cube. The dual of the cube is the octahedron, and its graph can also be realized as a Cayley graph. For this endeavor, we are assisted by the fact that there the octahe- Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 31 dron has only six vertices and there are only two groups of order 6, the cyclic group

Z6 = Z2 × Z3 and the dihedral group D6. How then, could each of these groups act on the octahedron freely and transitively on the vertices? For the cyclic group, the only candidate for an element of order 6 is a 60◦ rotary reﬂection interchanging two antipodal faces, see Figure 3.11, in which case the six (blue) edges which zigzag back

Figure 3.11: Two realizations of the octahedron graph as the Cayley graph of subgroups of isometries of the octahedron which act freely and transitively on the vertices. The left subgroup is isomorphic to Z6 and is generated by an order 6 rotary reﬂection about an axis through the centroids of an antipodal pair of triangular faces, with the second generator the square of this. The right subgroup is isomorphic to the dihedral group D6 and is generated by one rotation of order 3 about an axis through an antipodal face pair together with two rotations of order two along axes lying in the perpendicular plane and passing through the centers of antipodal edge pairs. and forth between those two faces are the connection between those two faces, and the red edges of those two faces themselves correspond to the square of the generat-

◦ ing rotary reflection, in other words, a rotation of 120 . For the dihedral group, D6, the elements of order three must be face rotations, so the three involutions must be rotations which interchange the two fixed faces. So D6 acts freely and transitively on the vertices of the graph of the octahedron, realizing it again as a Cayley graph. For the graph of the icosahedron, since the graph has valence 5, there must be at most one involutory generator, which by transitivity must be distributed as the red edges in Figure 3.12. These involutions must be rotations, since a reflection interchanging the endpoints of an edge does not act freely on all the vertices. Also, two edge rotations for incident edges generate the entire rotational group of the Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 32

Figure 3.12: A realization of the icosahedral graph as a Cayley graph for the alternating group acting a group of rotational symmetries of the cube generated by one 180◦ face rotations, two 120◦ vertex rotations around antipodal vertices. icosahedron which has 60 elements, far more than the 12 elements which we seek. So the remainder of the generators are not involutions, which must be divided up as the green and blue edges in the ﬁgure, realizing the graph of the icosahedron as the Cayley graph of a subgroup of the rotational group of the cube, isomorphic to the alternating group A4. Lastly we consider the graph of the dodecahedron, see Figure 3.13, which is

Figure 3.13: The dodecahedral graph has 120 automorphisms, but no subgroup of 20 elements acts freely and transitively on the vertices, and hence it is not a Cayley graph. Its quotient the Petersen graph also has 120 automorphisms but no 10 of them act freely and transitively on the vertices, and it is also not a Cayley graph.

3-valent and has 20 vertices. If there is one involutory generator, then the cycles of the non-involutory generator must either form four 5-cycles, five 4-cycles or two 10-cycles. But the dodecahedron has no four five-cycle, e.g. faces, which span the vertices. Nor does the dodecahedron have any 4-cycles, since the length of the short- est cycle is five. Lastly, any 10-cycle cuts the graph into two pieces, so the remainder of the vertices cannot be joined by a second 10-cycle. If there are only involutory Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 33 generators, we can consider any pair of them, they must partition the vertices of the graph into even cycles of the same length, but by the same considerations as above, this is impossible. So the dodecahedron is not the Cayley graph of any group, and neither is its relative, the Petersen graph. Chapter 4

Proof of the Main Theorem

In this chapter, we will present the proof of our main theorem which classiﬁes Cayley graphs over non-isomorphic abelian 2-groups. The proof reduces to a corollary, for which some background will be needed. In the ﬁrst section we provide the necessary background, and in the second section we present the proof.

4.1 W-subgroups of abelian groups

We introduce the following terminology:

Deﬁnition 4.1.1. Let G be an abelian group and S be a subset of G. We say that a subgroup H ≤ G is a W-subgroup relative to S if S \ H is a union of H-cosets.

We write H ≤S G when H is a W-subgroup relative to S, and also H

~~H ≤S G, the Cayley digraph X = Cay(G, S) can be decomposed into a wreath ∼ product X = X1 o X2, where X1 is a Cayley digraph on G/H and X2 is a Cayley digraph on H.~~

~~A chain G0 ≤ · · · ≤ Gm = G of subgroups of G we call a W-chain relative to S if~~

~~Gi ≤(S∩Gi+1) Gi+1 for all i ∈ {0, . . . , m − 1}.~~

~~Furthermore, we say that the chain is maximal if there exists no subgroup K ≤ G such that Gi <(S∩K) K <(S∩Gi+1) Gi+1 for some i ∈ {0, . . . , m − 1}.~~

~~34 Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 35~~

~~Proposition 4.1.2. Let G0 ≤ · · · ≤ Gm = G be a W -chain of an abelian group G relative to a subset S ⊆ G.~~

~~(i) For every i ∈ {0, . . . , m}, Gi ≤S G.~~

~~(ii) If Gb0 ≤ · · · ≤ Gbl = G0 is a W -chain of the subgroup G0 ≤ G relative to the~~

~~set S ∩ G0, then the concatenation~~

~~Gb0 ≤ · · · ≤ Gbl = G0 ≤ G1 ≤ · · · ≤ Gm = G~~

~~of the two W-chains is a W-chain of G relative to S.~~

~~(iii) Every cyclic group Zpn , p is a prime, has a unique maximal W-chain relative~~

~~to any subset S ⊆ Zpn .~~

~~Proof.~~

~~(i): The statement is true by deﬁnition for Gi = Gm. For an arbitrary subgroup~~

~~Gi, one may apply decreasing induction on the index i.~~

~~(ii): This follows immediately from the deﬁnition of a W-chain.~~

~~(iii): We prove this by induction on the exponent n. The statement is obvious if n = 1. Let n > 1, and assume that (iii) is true whenever the cyclic group has order~~

~~n0 n p < p . Set G = Zpn , and choose a maximal W-chain~~

~~1 = G0 < G1 < ··· < Gm = G~~

~~of G relative to S. There is a largest subgroup H < Zpn such that H~~

~~(Gm−1)S∩H ≤ H~~

The maximality of our W-chain gives that Gm−1 = H. The induction hypothesis implies that there is a unique maximal W-chain 1 = G0 < . . . < Gm−1 of Gm−1 relative to S ∩ Gm−1. Using (ii), this can be concatenated with Gm−1 < G to form the unique maximal chain of G relative to S, so (iii) follows.

As a consequence of Proposition 4.1.2 (iii) the following group is well-deﬁned. Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 36

~~Deﬁnition 4.1.3. For a subset S ⊆ Zpn , p is a prime, we deﬁne the group~~

~~W (Zpn ,S) = H1 × H2/H1 × · · · × Zpn /Hm−1,~~

~~where 1 = H0 < H1 < ··· < Hm−1 < Hm = Zpn is the unique maximal W-chain of~~

~~Zpn relative to S.~~

~~Let us turn back to Theorem 1.1.2. Following [16], we proceed by proving the implications: (1) ⇒ (2) ⇒ (3) ⇒ (1).~~

First, we notice that implications (2) ⇒ (3) and (3) ⇒ (1) can be shown by copying the argument of Morris (let us mention that implication (2) ⇒ (3) follows actually from a lemma of Joseph (see [10, Lemma 3.11]). For more details on these proofs the reader is advised to consult with the papers [10, 16]. The crucial part is implication (1) ⇒ (2), and this will follow as a corollary of the following theorem about maximal W-chains of Z2n , which we are going to prove in the next section.

~~Theorem 4.1.4. Let X = Cay(Z2n ,S), and~~

~~1 = H0 < H1 < ··· < Hm−1 < Hm = Z2n~~

~~be the unique maximal W-chain of Z2n relative to S. If X ∼= Cay(G, R) for some abelian group G, then there exists a W-chain~~

~~1 = G0 < G1 < ··· < Gm−1 < Gm = G~~

~~∼ of G relative to R so that Gi/Gi−1 = Hi/Hi−1 for all i ∈ {1, . . . , m}.~~

~~Corollary 4.1.5. Let X = Cay(G, R) be a Cayley digraph on an abelian group G~~

~~n of order 2 , and suppose that X is isomorphic to a Cayley digraph on both Z2n and H, where H is an abelian group with |H| = 2n (that is, the assumptions in (1) of Theorem 1.1.2 hold).~~

~~Then there exists a chain of subgroups G1 < ··· < Gm−1 in G such that~~

(i) G1, G2/G1, . . . , G/Gm−1 are cyclic groups, Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 37

~~(ii) G1 × G2/G1 × · · · × G/Gm−1 ≤po H,~~

~~(iii) for each 1 ≤ i ≤ m − 1, R \ Gi is a union of Gi-cosets.~~

~~∼ Proof. Let S be a subset of Z2n such that X = Cay(Z2n ,S), and let 1 = H0 <~~

~~H1 < ··· < Hm−1 < Hm = Z2n be the unique maximal W-chain of Z2n relative to S. By Theorem 4.1.4, there is a W-chain~~

~~1 = G0 < G1 < ··· < Gm−1 < Gm = G~~

~~∼ of G relative to R such that Gi/Gi−1 = Hi/Hi−1 for all i ∈ {1, . . . , m}. We claim that the chain G1 < ··· < Gm−1 satisﬁes all conditions (i)-(iii) in Corollary 4.1.5.~~

~~Condition (i) holds obviously, since H = Z2n is cyclic. Also, as Gi~~

~~∼ G1 × G2/G1 × · · · × G/Gm−1 = H1 × H2/H1 × · · · × Z2n /Hm−1 = W (Z2n ,S).~~

~~Therefore, W (Z2n ,S) ≤po G. We obtain in the same manner that W (Z2n ,S) ≤po H, which gives us condition (ii), and this completes the proof.~~

~~4.2 Proof of Theorem 4.1.4~~

We start with some notational comments. For the rest of the paper denote by Pi i the subgroup Pi ≤ Z2n of order 2 . Further, we set D2n+1 with n ≥ 2, and QD2n+1 with n ≥ 3 for the following overgroups of (Z2n )L in Sym(Z2n ):

~~D2n+1 = (Z2n )L, π1 , and QD2n+1 = (Z2n )L, π2 ,~~

−1 2n−1−1 where π1(x) = x for all x ∈ Z2n , and π2(x) = x for all x ∈ Z2n . The group n+1 D2n+1 is the dihedral group of order 2 , and the group QD2n+1 is also known as the quasidihedral group of order 2n+1 (see [9, Satz 14.9]).

Both permutations π1 and π2 normalize (Z2n )L. This implies that every subgroup of Z2n is a block of both permutation groups D2n+1 and QD2n+1 . The following property will be used later. Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 38

~~Proposition 4.2.1. If n ≥ 3 and G is a regular abelian subgroup in either D2n+1 or QD2n+1 , then G must be equal to (Z2n )L.~~

~~Next, we prove three lemmas about Cayley digraphs X = Cay(Z2n ,S) when the largest proper W-subgroup of Z2n relative to S has index greater than 2 in Z2n .~~

~~Lemma 4.2.2. Let X = Cay(Z2n ,S), and M be the largest proper subgroup of Z2n n−1 that M~~

~~(i) S contains a generator z of Z2n .~~

~~(ii) The orbit zAut(X)e is of the form:~~

~~Aut(X)e z = Pmz, m ∈ {0, . . . , n − 2}, or~~

~~n−1 Aut(X)e ε2 −1 z = Pmz ∪ Pmz , n ≥ 3, m ∈ {0, . . . , n − 3}, ε ∈ {0, 1}.~~

~~(iii) The group Pm in (ii) is a block of Aut(X).~~

~~Moreover, the induced permutation group (Aut(X))B is permutation isomor-~~

~~phic to (Z2n−m )L, or D2n−m+1 , or QD2n−m+1 , where B is the block system of~~

~~Aut(X) generated by Pm.~~

~~(iv) All subgroups Pi ≤ Z2n ,Pm ≤ Pi, are blocks of Aut(X).~~

~~Proof. We set A = Aut(X).~~

~~(i): If S contains no generator of Z2n then Pn−1~~

~~Ae (ii): Clearly, Cay(Z2n , z ) is connected and arc-transitive. The required form of~~

~~Ae Ae z follows from Theorem 2.3.1. Observe that, the case z = Z2n \Pm cannot arise.~~

~~Ae Since z ⊆ S and Z2n \ Pn−1 ⊆ Z2n \ Pm,Z2n \ Pm ⊆ S would give Pn−1~~

~~(iii): Apply Proposition 2.2.3 to the orbit zAe . From (i)~~

~~Ae Ae Pm = {x ∈ Z2n | xz = z },~~

~~by which Pm is a block of A. Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 39~~

~~zi−2 zi−1 zi zi+1 zi+2~~

~~zi+`−2 zi+`−1 zi+` zi+`+1 zi+`+2~~

~~Figure 4.1: The digraph Ce2n−m .~~

Identify the block system B with the factor group Z2n /Pm. We write x for the B B element Pmx in Z2n /Pm. Denote (A ) e the point stabilizer of e in the group A . B Because of (ii) the orbit of (A ) e generated by z equals:

~~{z}, or {z, z−1}, or {z, z2n−m−1−1}.~~

~~B ~ The respective orbital digraph of A is isomorphic to the directed cycle C2n−m , or the cycle C2n−m , or the Cayley digraph~~

`−1 Ce2n−m = Cay Z2n−m , {z, z } , where ` = 2n−m−1, see Figure 4.1. B ~ Moreover, the group A acts transitively on the arc set of the (di)graph C2n−m , ~ ∼ or C2n−m , or Ce2n−m , respectively. It is easily seen that Aut(C2n−m ) = (Z2n−m )L, ∼ Aut(C2n−m ) = D2n−m+1 , and Aut(Ce2n−m ) ≥ QD2n−m+1 , and each of these groups acts transitively on the arc set of the respective (di)graph. Therefore, (iii) follows if we prove that ∼ Aut(Ce2n−m ) = QD2n−m+1 . (4.1)

i i+` Let us consider the subsets {z , z } of vertices of Ce2n−m , see the picture of the graph drawn in Figure 4.1. These subsets are characterized by the property that their elements are joined in Ce2n−m by precisely two directed paths of length 2. Thus i i+` n−m the sets {z , z }, i ∈ {0, 1,..., 2 − 1}, induce a block system of Aut(Ce2n−m ). i i i+1 i+1 Now, choose g ∈ Aut(Ce2n−m ) such that g(z ) = z and g(z ) = z . It follows that g must ﬁx all blocks {zj, zj+`}, j ∈ {0, 1,..., 2n−m − 1}. Since (zi+1, zi+2) is

i+1 i+2 i+2+` the unique arc of Ce2n−m which starts at z and ends in the block {z , z }, we ﬁnd that g(zi+2) = zi+2. By the same reason g(zi+3) = zi+3 and, continuing on this manner, we deduce that g is the identity. Therefore, Aut(Ce2n−m ) acts semiregularly Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 40

~~on the set of arcs of Ce2n−m , and so by the orbit-stabilizer Lemma, | Aut(Ce2n−m )| ≤ n−m+1 2 . This together with Aut(Ce2n−m ) ≤ QD2n−m+1 yield (4.1).~~

~~B (iv): Because of (iii) all subgroups of Z2n /Pm are blocks of A . This implies precisely that all subgroups Pi ≤ Z2n ,Pm ≤ Pi, are blocks of A.~~

~~Lemma 4.2.3. Let X = Cay(Z2n ,S), and M be the largest proper subgroup of Z2n n−1 that M~~

~~(i) The orbit (z2) Aut(X)e is of the form:~~

~~2 Aut(X)e 2 (z ) = Plz , l ∈ {0, . . . , n − 2}, or~~

~~n−1 2 Aut(X)e 2 ε2 −2 (z ) = Plz ∪ Plz , n ≥ 4, l ∈ {0, . . . , n − 4}, ε ∈ {0, 1}.~~

~~(ii) All subgroups Pi ≤ Z2n ,Pl ≤ Pi, are blocks of Aut(X).~~

~~(iii) Let Pl be the subgroup given in (i) and Pm be the subgroup given in Lemma 4.2.2~~

~~(ii). Now, in case when Pl > Pm, we have Pm = Pn−3,~~

~~n−1 Aut(X)e ε2 −1 z = Pn−3z ∪ Pn−3z , n ≥ 3 and ε ∈ {0, 1}. (4.2)~~

~~2 Aut(X)e Furthermore, Pl = Pn−2, and so (z ) = Pn−1 \ Pn−2.~~

~~Proof. We set A = Aut(X).~~

~~(i): Let Pm be the subgroup given in Lemma 4.2.2 (i). As Pm ≤ Pn−2, both subgroups Pn−2 and Pn−1 are blocks of A, see Lemma 4.2.2 (iv). We may apply~~

~~2 Ae Theorem 2.3.1 to the digraph Cay Pn−1, (z ) to deduce (i). Observe that, the~~

~~2 Ae case (z ) = Pn−1 \ Pl, l < n − 2 cannot arise. Since Pn−2 is a block of A, the~~

~~2 A2 stabilizer Ae leaves Pn−2 setwise ﬁxed, implying that (z ) ∩ Pn−2 = ∅, so the case~~

~~Pn−1 \ Pl, l < n − 2 cannot arise.~~

~~Pn−1 (ii): Consider the permutation group Q = (A{Pn−1}) in Sym(Pn−1). Since~~

Pn−1 is a block of A, every block of Q is a block of A as well, and thus it suﬃces to show that the subgroups Pi ≤ Pn−1,Pl ≤ Pi, are blocks of Q. Let Qe denote the point stabilizer of e in Q. Clearly, the orbit (z2) Qe = (z2) Ae . Apply (iii) and (iv) Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 41

~~in Lemma 4.2.2 to get that all subgroups Pi ≤ Pn−1,Pl ≤ Pi, are blocks of Q. This implies (ii).~~

~~(iii): Let us assume that Pl > Pm. By (i),~~

~~ 2 Ae 2 Ae Pl = x ∈ Z2n | (z ) x = (z ) .~~

~~Let Pm be the subgroup given in Lemma 4.2.2 (i). As previously, let B denote the block system generated by the block Pm, which is identiﬁed by the factor group~~

~~Z2n /Pm, and write x for Pmx ∈ Z2n /Pm.~~

~~B 2 (A ) e 2 B ∼ Using Lemma 4.2.2 (iii) we see that the orbit (z ) = {z } if A = Z2n−m , and~~

~~B B 2 (A ) e 2 −2 B ∼ B ∼ 2 Ae 2 (A ) e (z ) = {z , z } if A = D2n−m+1 or A = QD2n−m+1 . Then (z ) = (z ) . Therefore,~~

~~B B 2 (A ) e 2 (A ) e P l = x ∈ Z2n /Pm | (z ) x = (z ) .~~

~~B 2 (A ) e 2 −2 2 −2 Thus unless (z ) = {z , z } and the set {z , z } is a coset in Z2n /Pm, P l = 1, which implies Pl ≤ Pm, a contradiction.~~

~~B 2 (A ) e 2 −2 2 Hence (z ) = {z , z } and this is a coset in Z2n /Pm. The element z n−m−1 2 −2 4 has order 2 in Z2n /Pm. Since {z , z } is a coset, z has order 2 in Z2n /Pm.~~

~~Ae These give us n − m = 3, i.e., Pm = Pn−3. Now, (4.2) follows unless z = Pn−3z.~~

B B ∼ 2 (A ) e 2 In the latter case, however, we have A = Z8, from which (z ) = {z }. This ∼ ∼ is a contradiction, hence (4.2) holds. Furthermore, Z2n /Pm = Z8 and Pl/Pm = Z2, giving us that Pl = Pn−2. This completes the proof of (iii).

~~Lemma 4.2.4. Let X = Cay(Z2n ,S), and M be the largest proper subgroup of Z2n n−1 that M~~

~~Proof. We prove the lemma by induction on n. The statement is trivial if n = 2. 0 Let n > 2 and assume that the lemma holds for any Cayley digraph on Z2n0 if n < n. Set A = Aut(X).~~

~~By Lemma 4.2.2 (i) the set S contains a generator z of Z2n . First we consider the orbits zAe and (z2) Ae . By Lemma 4.2.2 (i),~~

n−1 Ae ε2 −1 z = Pmz ∪ ( ∆ ∩ Pmz ) (4.3) Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 42

~~for suitable numbers m ∈ {0, . . . , n − 2}, ε ∈ {0, 1}, and set ∆ ∈ {∅, Z2n }. If~~

~~M ≥ Pm then M is a block because of Lemma 4.2.2 (iv), hence we assume that~~

~~M < Pm. By Lemma 4.2.3 (i),~~

~~n−1 2 Ae 2 ε2 −2 (z ) = Plz ∪ ( ∆ ∩ Plz ) (4.4)~~

for suitable numbers l ∈ {0, . . . , n − 3}, ε ∈ {0, 1}, and set ∆ ∈ {∅, Z2n },. If M ≥ Pl then M is a block because of Lemma 4.2.3 (iii), hence we assume that M < Pl. In view of Theorem 2.2.4 the set S \ Pn−1 is described as

~~Ae Ae (k1) Ae (kr) S \ Pn−1 = z ∪ z ∪ · · · ∪ z (4.5)~~

~~for some odd numbers k1, . . . , kr. Furthermore, if S ∩ (Pn−1 \ Pn−2) 6= ∅, then~~

~~2 Ae (l1) 2 Ae (ls) S ∩ (Pn−1 \ Pn−2) = (z ) ∪ · · · ∪ (z ) (4.6)~~

~~for suitable odd numbers l1, . . . , ls.~~

~~We deﬁne two auxiliary Cayley digraphs which depend on X:~~

~~2 Ae X1 = Cay(Pn−1,S1), where S1 = (S ∩ Pn−2) ∪ (z ) . (4.7)~~

~~X2 = Cay(Pn−2,S2), where S2 = S ∩ Pn−2. (4.8)~~

Notice that since both Pn−1 and Pn−2 are blocks of A, the sets S1 and S2 are ﬁxed by Ae. Thus for both i = 1, 2, every element in the set-wise stabilizer A{Pn−i} acts on Pn−i as an automorphism of Xi. Therefore,

~~Pn−i (A{Pn−i}) ≤ Aut(Xi) for both i = 1, 2.~~

~~We split the argument into two cases according to whether (4.2) holds or not.~~

~~n−1 Ae ε2 −1 CASE 1. z 6= Pn−3z ∪ Pn−3z or n < 3 or ε 6∈ {0, 1}.~~

~~Since M < Pl ≤ Pn−2, (4.4) and (4.7) show that M~~

~~By Lemma 4.2.3 (iii), Pl ≤ Pm. Using the deﬁnition of S1, we obtain from~~

~~K ≤ Pl ≤ Pm,S \ Pn−2 is a union of K-cosets as well. This implies together with~~

~~n n K ≤S∩Pn−2 Pn−2 that K~~

~~n−2 in Pn−1 such that M~~

~~Pn−1 Pn−1 Aut(X1). As (A{Pn−1}) ≤ Aut(X1), M is also a block of (A{Pn−1}) . This implies that M is a block of A, as claimed.~~

~~n−1 Ae ε2 −1 CASE 2. z = Pn−3z ∪ Pn−3z , n ≥ 3 and ε ∈ {0, 1}.~~

~~Recall that M < Pm = Pn−3. Let K be a subgroup of Pn−2 such that M ≤~~

~~2 A2 K < Pn−2 and K~~

Pn−1\Pn−2, in particular, S∩(Pn−1\Pn−2) is a union of Pn−3-cosets. Notice also that, by (4.5), S \ Pn−1 is a union of Pn−3-cosets. As K ≤ Pn−3 and K ≤S∩Pn−2 Pn−2, we conclude that K

~~Pn−2 Therefore M is a block of (A{Pn−2}) ≤ Aut(X2), and so a block of A as well.~~

~~We are going to prove Theorem 4.1.4 by induction on the length m of the unique maximal W-chain 1 = H0 < H1 < ··· < Hm−1 < Hm = Z2n . The basic case m = 1 is done in the following lemma.~~

~~Lemma 4.2.5. Let X = Cay(Z2n ,S), and suppose that there is no subgroup of~~

~~1 < H < Z2n such that H~~

Proof. We prove the lemma by induction on n. If n ≤ 3, then the statement can be veriﬁed by hand. Thus we assume that n ≥ 4. We set A = Aut(X) and let G denote a regular abelian subgroup of A. Let us keep the setting from the previous proof:

~~1. S contains a generator z of Z2n (by Lemma 4.2.2 (i)),~~

~~Ae 2. the orbit z is given by (4.3), and the set S \ Pn−1 by (4.5),~~

2 Ae 3. the orbit (z ) is given by (4.4), and the set S ∩ (Pn−1 \ Pn−2) by (4.6), Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 44

~~4. X1 and X2 are the auxiliary Cayley digraphs deﬁned according to (4.7) and (4.8), respectively.~~

The subgroup Pm is a block of A because of Lemma 4.2.2 (iii). Denote by B the block B system of A consisting of the Pm-cosets in Z2n . Then, still by Lemma 4.2.2 (iii), A is permutation isomorphic to Z2n−m , or D2n−m+1 , or QD2n−m+1 . Recall that G is an B ∼ abelian regular subgroup G ≤ A. By Proposition 2.2.1, the group G = GAB/AB B is an abelian regular subgroup of A , where AB is the kernel of A acting on B. We claim that ∼ B ∼ GAB/AB = G = Z2n−m . (4.9)

~~B ∼ B ∼ This is obviously true if A = Z2n−m . Otherwise, A = D2n−m+1 or QD2n−m+1 , and in each of these cases, by Lemma 4.2.2 (ii), m ≤ n − 3. Then (4.9) follows from Proposition 4.2.1.~~

~~CASE 1. There is no subgroup 1 < K < Pn−1 such that K~~

~~Let H = G ∩ A{Pn−1}. Then H is regular on Pn−1, and~~

~~∼ Pn−1 Pn−1 H = H ≤ (A{Pn−1}) ≤ Aut(X1).~~

∼ ∼ Induction gives H = Z2n−1 . Now, if G 6= Z2n , then there exists an involution g ∈ G 0 0 ∼ n−1 such that G = H × hgi = Z2 × Z2. Let g ∈ G ∩ AB. Then g ∈ A{Pn−1}, and it follows G ∩ AB ≤ H. Thus hgi ∩ (G ∩ AB) = 1, and so

~~∼ ∼ ∼ GAB/AB = G/(G ∩ AB) = H/(G ∩ AB) × hgi/(hgi ∩ G ∩ AB) = Z2n−1−m × Z2.~~

~~∼ This contradicts (4.9), hence G = Z2n .~~

~~CASE 2. There exists a subgroup 1 < K < Pn−1 such that K~~

~~We claim that in this case (4.2) holds, i.e.,~~

~~n−1 Ae ε2 −1 z = Pn−3z ∪ Pn−3z , n ≥ 3 and ε ∈ {0, 1}.~~

~~Indeed, otherwise Pl ≤ Pm because of Lemma 4.2.3 (iii). Using the deﬁnition of~~

~~S1, we obtain from K~~

(4.5), S \ Pn is a union of Pm-cosets, and by (4.6), S ∩ (Pn−1 \ Pn−2) is a union of Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 45

~~Pl-cosets. As K ≤ Pl ≤ Pm,S \ Pn−2 is a union of K-cosets, which implies together~~

~~n with K ≤S∩Pn−2 Pn−2 that 1 < K~~

~~Let K be a subgroup 1 ≤ K < Pn−2 such that K~~

~~2 Ae (z ) = Pn−1 \ Pn−2 holds in this case. We conclude that there is no subgroup~~

~~1 < K < Pn−2 such that K~~

~~Let H = G ∩ A{Pn−2}. Then H is regular on Pn−2, and~~

~~∼ Pn−2 Pn−2 H = H ≤ (A{Pn−2}) ≤ Aut(X2).~~

~~∼ ∼ Thus H = Z2n−2 . If G 6= Z2n , then there exists g ∈ G, such that H ∩ hgi = 1. Using~~

~~Pm = Pn−3 < Pn−2, it follows readily that G ∩ AB < H. Thus hgiAB/AB 6= 1 and~~

~~HAB/AB 6= 1, implying that GAB/AB cannot be a cyclic group. This contradicts ∼ (4.9), hence G = Z2n .~~

~~Proof of Theorem 4.1.4. We prove the theorem by induction on the length m of the maximal W-chain~~

~~1 = H0 < H1 < ··· Hm−1 < Hm = Z2n (4.10)~~

of Z2n relative to S. Let m = 1. This means that there exists no subgroup 1 < K < ∼ Z2n such that K ~~1, and assume that the theorem holds for any X = Cay(Z2n0 ,S ) such 0 0 that Z2n0 has maximal W-chain relative to S of length m < m.~~

~~CASE 1. Hm−1 < Pn−1.~~

By Lemma 4.2.4, Hm−1 is a block of Aut(X). Consider the quotient Cayley digraph (see 2.2) ∼ X/Hm−1 = Cay(Z2n /Hm−1, S/Hm−1). Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 46

~~Because of the maximality of the W-chain in (4.10) there exists no subgroup 1 <~~

~~n n K < Z2 /Hm−1 with K~~

~~∼ Y [D1; D2] = Kn0,n0 . (4.11)~~

Here Y [D1; D2] is the bipartite subgraph of Y induced by the sets D1 and D2, that is, it has vertex set D1 ∪ D2, and {x1, x2} is an edge of Y [D1; D2] if and only if x1 ∈ D1, x2 ∈ D2 and {x1, x2} is an edge of Y . By Corollary 2.2.2, D is formed by the cosets of a subgroup L ≤ G, |L| = n0. Let ∼ us set Gm−1 to be L. Property (4.11) implies that Gm−1

~~Y/Gm−1, and Aut(X/Hm−1) has a regular subgroup isomorphic to Gm/Gm−1. We ∼ obtain that Gm/Gm−1 = Z2n /Hm−1 = Hm/Hm−1. Finally, consider the Cayley digraphs~~

~~X1 = Cay(Hm−1,S ∩ Hm−1) and Y1 = Cay(Gm−1,R ∩ Gm−1).~~

∼ It follows that X1 = Y1. Also, 1 = H0 < H1 < ··· < Hm−1 is the maximal W-chain of Hm−1 relative to S ∩ Hm−1 which is of length m − 1 (see Proposition 4.1.2). By induction there exists a W-chain 1 = G0 < G1 < ··· < Gm−1 of Gm−1 relative to ∼ R ∩ Gm−1 such that Gi/Gi−1 = Hi/Hi−1 for every i ∈ {1, . . . , m − 1}.

~~Now, the concatenation 1 = G0 < G1 < ··· < Gm−1 < Gm = Z2n is a W-chain ∼ of G relative to R by Proposition 4.1.2 (ii). Further, Gi/Gi−1 = Hi/Hi−1 for all i ∈ {1, . . . , m}, as required.~~

~~CASE 2. Hm−1 = Pn−1.~~

~~Ae Fix a generator z of Z2n . In this case the orbit z = Z2n \ H for some subgroup~~

~~H < Z2n . Therefore (4.10) is rewritten as~~

1 = H0 < H1 < ··· < H = H` < H`+1 ··· < Hm−1 < Hm = Z2n , where ` ∈ {0, . . . , m − 1}, and [Hi : Hi−1] = 2 for all i ∈ {` + 1, . . . , m}. Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 47

~~It is easy to see that H is a block of Aut(X). Then the quotient Cayley digraph~~

~~∼ n n ∼ X/H = Cay(Z2 /H, Z2 /H \ H) = K|Z2n /H|.~~

~~Set Y = Cay(G, R). Because of X ∼= Y , there is a subgroup L ≤ G, |L| = |H|, such that L is a block of Aut(Y ), and~~

~~∼ ∼ Y/L = X/H = K|Z2n /H|.~~

~~Let us set G` to be L, and choose a chain~~

~~G` < G`+1 < ··· < Gm−1 < Gm = G~~

of subgroups of G such that [Gi : Gi−1] = 2 for all i ∈ {` + 1, . . . , m}. Obviously, ∼ ∼ this is a W-chain of G relative to R, and Gi/Gi−1 = Hi/Hi−1 = Z2 for every i ∈ {` + 1, . . . , m}. Finally, we need to consider the Cayley digraphs

~~X1 = Cay(H`,S ∩ H`) and Y1 = Cay(G`,R ∩ G`).~~

~~∼ It follows that X1 = Y1. Also, 1 = H0 < H1 < ··· < H` is the maximal W-chain of~~

H` relative to S ∩ H` which is of length m − `. By induction there exists a W-chain ∼ 1 = G0 < G1 < ··· < G` of G` relative to R ∩ G` such that Gi/Gi−1 = Hi/Hi−1 for every i ∈ {1, . . . , `}. Now, the required W-chain of G is obtained as the concatenation of the W-chain

~~1 = G0 < G1 < ··· < G` of G` relative to R ∩ G` with the W-chain G` < G`+1 <~~

~~··· < Gm−1 < Gm = G of G relative to R. This completes the proof. Chapter 5~~

~~Conclusions~~

~~In this thesis we have considered the question of which graphs can be represented as Cayley graphs over non-isomorphic groups. We restricted our work to three special classes of graphs:~~

~~(1) Several examples of infinite graphs represented as Cayley graphs of different infinite groups.~~

~~(2) The graphs of the ﬁve Platonic solids.~~

~~(3) Circulants (Cayley graphs over cyclic groups) of 2-power order.~~

In case (1) we studied infinite Cayley graphs which arise from infinite treelike graphs and the plane lattices (square, triangular, and hexagonal). With each graph we associated at least two groups defined by a suitable presentation, showed that each of these groups induces a Cayley representation of the given graph, and that they are non-isomorphic. In some cases we also provided a geometric interpretation of the group.

In part (2) we tackled a classical problem. It is well-known that there are five Platonic solids, and the full symmetry group of each solid (including reflections) is isomorphic to the automorphism group of the corresponding graph. This allows us to find Cayley representations of the graphs by simply searching for symmetry groups acting freely and transitively on the vertices. We carried out this analysis

48 Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 49 for each Platonic solid; in particular, gave an elementary argument why the graph of the cube has exactly 10 Cayley representations.

In part (3) we considered a research problem. In the paper [16], Morris gave a nice answer for the initial question on Cayley representations in the class of circulants of odd prime-power order. She also pointed out that the same answer holds for circulants of 2-power order which are at the same time represented as Cayley graphs over elementary abelian 2-groups. We managed to prove that Morris’s theorem holds also in the case when one replaces the elementary abelian 2-group with an arbitrary abelian non-cyclic 2-group. Our result was published in J. Graph Theory 70 (2012), 435–448. To conclude this summary, and the thesis as well, it is worth to mention that Mor- ris’s theorem was generalized very recently by Dobson and Morris [5] to circulants of arbitrary order which can be represented as Cayley graphs over some non-cyclic abelian group. The problem of how to generalize it to non-abelian groups is still wide open. One of the ﬁrst cases one may try to attack is the class of circulants of 2-power order. Bibliography

~~[1]A. Ad´am.Research´ problem 2. J. Combinatorial Theory, 2:309, 1967.~~

~~[2] F. Buckley and M. Lewinter. A Friendly Introduction to Graph Theory. Pearson Education, Prentice Hall, 2003.~~

~~[3] Nathan C. Carter. Visual group theory. Classroom Resource Materials Series. Mathematical Association of America, Washington, DC, 2009.~~

~~[4] John D. Dixon and Brian Mortimer. Permutation groups, volume 163 of Grad- uate Texts in Mathematics. Springer-Verlag, New York, 1996.~~

~~[5] J. Morris E. Dobson. On circulant graphs that are isomorphic to cayley graphs of more than one abelian group. Modern Trends in Algebraic Graph Theory, an International Conference, 2014.~~

~~[6] Bernard Elspas and James Turner. Graphs with circulant adjacency matrices. J. Combinatorial Theory, 9:297–307, 1970.~~

~~[7] John B. Fraleigh. A ﬁrst course in abstract algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967.~~

~~[8] Chris Godsil and Gordon Royle. Algebraic graph theory, volume 207 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2001.~~

~~[9] B. Huppert. Endliche Gruppen. I. Die Grundlehren der Mathematischen Wis- senschaften, Band 134. Springer-Verlag, Berlin, 1967.~~

~~[10] Anne Joseph. The isomorphism problem for Cayley digraphs on groups of prime-squared order. Discrete Math., 141(1-3):173–183, 1995.~~

~~50 Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 51~~

~~[11] Istv´anKov´acs.Classifying arc-transitive circulants. J. Algebraic Comb., 20:353– 358, November 2004.~~

~~[12] Cai Heng Li. On isomorphisms of ﬁnite Cayley graphs—a survey. Discrete Math., 256(1-2):301–334, 2002.~~

~~[13] Cai Heng Li. The ﬁnite primitive permutation groups containing an abelian regular subgroup. Proc. London Math. Soc. (3), 87(3):725–747, 2003.~~

~~[14] Cai Heng Li. Permutation groups with a cyclic regular subgroup and arc transitive circulants. J. Algebraic Combin., 21(2):131–136, 2005.~~

~~[15] LászlóLovász. Combinatorial problems and exercises. North-Holland Publish- ing Co., Amsterdam, second edition, 1993.~~

~~[16] Joy Morris. Isomorphic cayley graphs on nonisomorphic groups. J. Graph Theory, 31:345–362, August 1999.~~

~~[17] Mikhail E. Muzychuk. On the structure of basic sets of Schur rings over cyclic groups. J. Algebra, 169(2):655–678, 1994.~~

~~[18] Helmut Wielandt. Finite permutation groups. Translated from the German by R. Bercov. Academic Press, New York, 1964.~~