Orders, Conjugacy Classes, and Coverings of Permutation Groups by Attila Mar¶oti A thesis to be submitted to the University of Szeged for the degree of Ph. D. in the Faculty of Sciences August 2007 Acknowledgements I thank my supervisor, Professor L¶aszl¶oPyber for his generous help, patience, and guidance. Thanks are also due to Professors Agnes¶ Szendrei, B¶alintn¶eSzendrei M¶aria,L¶aszl¶o Hatvani, and to Edit Annus. Contents 1 Introduction 1 2 Preliminaries 3 2.1 Basic de¯nitions . 3 2.2 Blocks and primitivity . 4 2.3 The Aschbacher-O'Nan-Scott theorem . 5 3 On the orders of primitive permutation groups 9 3.1 Introduction . 9 3.2 Proof of Theorem 3.1.1. 13 3.3 Corollaries . 17 3.4 An application . 19 4 On the number of conjugacy classes of a permutation group 20 4.1 Introduction . 20 4.2 Linear groups . 22 4.3 Primitive permutation groups . 28 4.4 The general bound . 36 4.5 Groups with no composition factor of order 2 . 39 4.6 Nilpotent groups . 42 5 Covering the symmetric groups with proper subgroups 46 2 5.1 Introduction . 46 5.2 Preliminaries . 49 5.3 Symmetric groups . 49 5.4 Alternating groups . 56 5.5 A Mathieu group . 62 5.6 On some in¯nite series of σ ........................ 64 5.7 An application . 65 6 Summary 67 7 Osszefoglal¶oÄ 71 Chapter 1 Introduction Permutation groups arguably form the oldest part of group theory. Their study dates back to the early years of the nineteenth century and, indeed, for a long time groups were always understood to be permutation groups. Although, of course, this is no longer true, permutation groups continue to play an important role in modern group theory through the ubiquity of group actions and the concrete representations which permutation groups provide for abstract groups. This thesis is built around three papers of the author; all three involving permuta- tion groups. Chapter 3 considers a very old problem going back to Jordan and Bochert of bounding the order of a primitive permutation group of degree n not containing the alternating group An. This chapter is taken from [45]. Chapter 4 is an early version of the paper [48]. Here we consider the problem of bounding the number of conjugacy classes of a permutation group of degree n > 2. This problem is related to the so- called k(GV )-problem of group theory and more distantly to Brauer's k(B)-problem of representation theory. The previous problem was solved recently, while the latter is unsolved. The results in Chapter 4 will be used in [28] where we consider the so-called non-coprime k(GV )-problem proposed by Guralnick and Tiep in 2004. Finally, Chap- ter 5 deals with a more combinatorial problem of covering the symmetric groups by proper subgroups. This material is taken directly from [47]. 1 Chapter 2 contains the preliminaries to this thesis. Not everything in Section 2.4 is used later in the text. For more background the reader can use [25] and [20]. Chapter 2 Preliminaries 2.1 Basic de¯nitions Let ­ be an arbitrary nonempty set; its elements are often called points. A bijection of ­ onto itself is called a permutation of ­. The set of all permutations of ­ forms a group, under composition of mappings, called the symmetric group on ­. This group is denoted by Sym(­). If ­ = f1; 2; : : : ; ng for some positive integer n, then Sym(­) is abbreviated as Sn. A subgroup of the symmetric group (on ­) is called a permutation group (on ­). If G and H are permutation groups on ­ and ¢, respectively, then we say that G is permutation equivalent to H if there is a bijection Á : ­ ! ¢ and an isomorphism à : G ! H such that (!g)Á = (!Á)(gÃ) for all g 2 G, ! 2 ­. Let G be any group and ­ be any nonempty set. Suppose we have a function from ­ £ G into ­ such that the image of a pair (®; x) is denoted by ®x for every ® 2 ­ and x 2 G. We say that this function de¯nes an action of G on ­ (or we say that G acts on ­) if the following two conditions hold. (i) ®1 = ® for all ® 2 ­; (ii) (®x)y = ®xy for all ® 2 ­ and all x, y 2 G. For example, if H is any subgroup of any group G, then G acts on the set of right 3 cosets of H in G in a natural way. When a group G acts on a set ­, a typical point ® is moved by elements of G to various other points. The set of these images is called the orbit of ® under G (or the G-orbit containing ®), and we denote it by ®G = f®x j x 2 Gg. A kind of dual role is played by the set of elements in G which ¯x a speci¯ed point ®. This is called the x stabilizer of ® in G and is denoted G® = fx 2 G j ® = ®g. Suppose that G is a group acting on a set ­ and that x, y 2 G and ®, ¯ 2 ­. Then the following three statements are true. (i) Two orbits ®G and ¯G are either equal (as sets) or disjoint, so the set of all orbits is a partition of ­ into mutually disjoint subsets. ¡1 x (ii) The stabilizer G® is a subgroup of G and G¯ = x G®x whenever ¯ = ® . x y Moreover, ® = ® if and only if G®x = G®y. G (iii) We have j® j = jG : G®j for all ® 2 ­. In particular, if G is ¯nite then G j® jjG®j = jGj. A group G acting on a set ­ is said to be transitive on ­ if it has only one orbit, and so ®G = ­ for all ® 2 ­. Equivalently, G is transitive if for every pair of points ®, ¯ 2 ­ there exists x 2 G such that ®x = ¯. A group which is not transitive is called intransitive. A group G acting transitively on a set ­ is said to act regularly if G® = f1g for each ® 2 ­ (equivalently, only the identity ¯xes any point). Similarly, a permutation group G is called regular if it is transitive and only the identity ¯xes any point. 2.2 Blocks and primitivity In what follows we shall extend the action of G on ­ to subsets of ­ by de¯ning ¡x = fγx j γ 2 ¡g for each ¡ ⊆ ­. Let G be a group acting transitively on a set ­. A nonempty subset ¢ of ­ is called a block for G if for each x 2 G either ¢x = ¢ or ¢x \¢ = ;. Every group acting transitively on ­ has ­ and the singletons f®g (® 2 ­) as blocks. These two types of blocks are called trivial blocks. Any other block is called nontrivial. The importance of blocks arises from the following observation. Suppose that G acts transitively on ­ and that ¢ is a block for G. Put § = f¢x j x 2 Gg. Then the sets in § form a partition of ­ and each element of § is a block for G. We call § the system of blocks containing ¢. Now G acts on § in an obvious way, and this new action may give useful information about G provided ¢ is not a trivial block. Let G be a group acting transitively on a set ­. We say that the group is primitive if G has no nontrivial block on ­. Otherwise G is called imprimitive. We may also talk about primitive and imprimitive permutation groups. Let G act on a set ­. This action may be transitive or intransitive. If it is transitive, then it can be primitive or imprimitive. We will mainly be interested in primitive groups. Hence it is useful to mention the following fact. The transitive group G is primitive if and only if the point stabilizer G® is a maximal subgroup in G for all ® 2 ­. 2.3 The Aschbacher-O'Nan-Scott theorem The so-called Aschbacher-O'Nan-Scott theorem gives a description of ¯nite primitive permutation groups (primitive permutation groups with ¯nitely many elements). This section closely follows the paper of Liebeck, Praeger, Saxl [41]. For more information on individual groups the reader may use [20] as a reference. A minimal normal subgroup of a nontrivial group X is a normal subgroup K 6= f1g of X which does not contain properly any other nontrivial normal subgroup of X. For example, a simple group itself is its only minimal normal subgroup, while an in¯nite cyclic group has no minimal normal subgroup. The socle of a group X is the subgroup generated by the set of all minimal normal subgroups of X. It is denoted by soc(X). By convention, we put soc(X) = f1g if X has no minimal normal subgroup. Since the set of all minimal normal subgroups of X is mapped into itself by every automorphism of X, the socle soc(X) is a characteristic subgroup of X. Every nontrivial ¯nite group has at least one minimal normal subgroup so has a nontrivial socle. For groups A and B we denote by A:B a (not necessarily split) extension of A and B. The split extension of A by B is denoted by A : B. In what follows, X will be a primitive permutation group on a ¯nite set ­ of size n, and ® a point in ­. Let B be the socle of X. The socle of a ¯nite primitive permutation group is the direct product of a simple group. So in this case, B »= T k for some simple group T and some integer k ¸ 1. Consider the following types of permutation groups: I, II, III(a), III(b), and III(c).