On minimal subdegrees of finite primitive permutation groups

We give a new characterization of primitive permutation groups tied to the notion of completely reachable automata. Also, we introduce sync-maximal permutation groups tied to the state complexity of the set of synchronizing words of certain associated automata and show that they are contained between the 2-homogeneous and the primitive groups. Lastly, we define k-reachable groups in analogy with synchronizing groups and motivated by our characterization of primitive permutation groups. But the results show that a k-reachable permutation group of degree n with \(6 \le k \le n - 6\) is either the alternating or the symmetric group.

This thesis records an attempt to prove the two conjecture: Conjecture A: Every finite non-regular primitive permutation group of degree n contains permutations fixing one point but fixing at most $n^{1/2}$ points. Conjecture C: Every finite irreducible linear group of degree m > 1 contains an element whose fixed-point space has dimension at most m/2. Variants of these conjectures are formulated, and C is reduced to a special case of A. The main results of the investigation are: Theorem 2: Every finite non-regular primitive permutation group of degree n contains permutations which fix one point but fix fewer than (n+3)/4 points. Theorem 3: Every finite non-regular primitive soluble permutation group of degree n contains permutations which fix one point but fix fewer than $n^{7/18}$ points. Theorem 4: If H is a finite group, F is a field whose characteristic is 0 or does not divide the order of H, and M is a non-trivial irreducible H-module of dimension m over F, then there is an element h in H whose fixed-point space in M has dimension less than m/2. Theorem 5: If H is a finite soluble group, F is any field, and M is a non-trivial irreducible H-module of dimension m over F, then there is an element h in H whose fixed-point space in M has dimension less than 7m/18. Proofs of these assertions are to be found in Chapter II; examples which show the limitations on possible strenghtenings of the conjectures and results are marshalled in Chapter III. A detailed formulation of the problems and results is contained in section 1.

First we define and study the exponentiation of a cellular algebra by a permutation group which is analogous to the corresponding operation (the wreath product in primitive action) in permutation group theory. Necessary and sufficient conditions for the resulting cellular algebra to be primitive and Schurian are given. This enables us to construct infinite series of primitive non-Schurian algebras. Also we define and study for cellular algebras the notion of a base which is similar to that for permutation groups. We present an upper bound for the size of an irredundant base of a primitive cellular algebra in terms of the parameters of its standard representation. This produces new upper bounds for the order of the automorphism group of such an algebra and in particular for the order of a primitive permutation group. Finally we generalize to 2-closed primitive algebras some classical theorems for primitive groups and show that the hypothesis for a primitive algebra to be 2-closed is essential.

Generation and random generation: From simple groups to maximal subgroups

This is the first in a series of papers whose results imply the validity of a strengthened version of the Sims conjecture on finite primitive permutation groups from the authors’ article “Stabilizers of graph’s vertices and a strengthened version of the Sims conjecture”, Dokl. Math. 59 (1), 113–115 (1999). In this paper, the case of not almost simple primitive groups and the case of primitive groups with alternating socle are considered.

A permutation group $(X,G)$ is said to be binary, or of relational complexity $2$, if for all $n$, the orbits of $G$ (acting diagonally) on $X^2$ determine the orbits of $G$ on $X^n$ in the following sense: for all $\bar{x},\bar{y} \in X^n$, $\bar{x}$ and $\bar{y}$ are $G$-conjugate if and only if every pair of entries from $\bar{x}$ is $G$-conjugate to the corresponding pair from $\bar{y}$. Cherlin has conjectured that the only finite primitive binary permutation groups are $S_n$, groups of prime order, and affine orthogonal groups $V\rtimes O(V)$ where $V$ is a vector space equipped with an anisotropic quadratic form; recently he succeeded in establishing the conjecture for those groups with an abelian socle. In this note, we show that what remains of the conjecture reduces, via the O'Nan-Scott Theorem, to groups with a nonabelian simple socle.

A permutation group G ≤ Sym(X) on a finite set X is sharp if |G|=∏ l∈L(G)(|X| − l), where L(G) = {|fix(g)| | 1 ≠ g ∈ G}. We show that no finite primitive permutation groups of twisted wreath type are sharp.

Primitive permutation groups with a solvable 2-transitive subconstituent

On the orders of primitive groups

Primitive permutation IBIS groups

Invariant theory for wreath product groups

J.M. Howie, the influential St Andrews semigroupist, claimed that we value an area of pure mathematics to the extent that (a) it gives rise to arguments that are deep and elegant, and (b) it has interesting interconnections with other parts of pure mathematics. This paper surveys some recent results on the transformation semigroup generated by a permutation group $G$ and a single non-permutation $a$. Our particular concern is the influence that properties of $G$ (related to homogeneity, transitivity and primitivity) have on the structure of the semigroup. In the first part of the paper, we consider properties of $S= $ such as regularity and idempotent generation. The second is a brief report on the synchronization project, which aims to decide in what circumstances $S$ contains an element of rank 1. The paper closes with a list of open problems on permutation groups and linear groups, and some comments about the impact on semigroups are provided. These two research directions outlined above lead to very interesting and challenging problems on primitive permutation groups whose solutions require combining results from several different areas of mathematics, certainly fulfilling both of Howie's elegance and value tests in a new and fascinating way.

Let G be a transitive permutation group on a set Ω such that, for ω∈Ω, the stabiliser Gω induces on each of its orbits in Ω∖{ω} a primitive permutation group (possibly of degree 1). Let N be the normal closure of Gω in G. Then (Theorem 1) either N factorises as N=GωGδ for some ω, δ∈Ω, or all unfaithful Gω-orbits, if any exist, are infinite. This result generalises a theorem of I. M. Isaacs which deals with the case where there is a finite upper bound on the lengths of the Gω-orbits. Several further results are proved about the structure of G as a permutation group, focussing in particular on the nature of certain G-invariant partitions of Ω. 1991 Mathematics Subject Classification 20B07, 20B05.

This paper surveys some results in the area of maximal subgroups of the finite simple groups and their automorphism groups. The first two sections are concerned with maximal subgroups of the alternating and symmetric groups. We outline the Reduction Theorem and discuss the maximality of primitive groups in the corresponding symmetric groups. In the third section we consider the corresponding questions for the classical groups. The next section contains a brief survey of knowledge of the maximal subgroups of the other almost simple groups. Finally, the last section outlines some recent work on primitive permutation groups of special degrees.

The structure of the commutator subgroup of Sylow 2-subgroups of an alternating group A 2 k is determined. This work continues the previous investigations of me, where minimal generating sets for Sylow 2-subgroups of alternating groups were constructed. Here we study the commutator subgroup of these groups. The minimal generating set of the commutator subgroup of A 2 k is constructed. It is shown that ( S y l 2 A 2 k ) 2 = S y l 2 ′ A 2 k , k > 2 . It serves to solve quadratic equations in this group, as were solved by Lysenok I. in the Grigorchuk group. It is proved that the commutator length of an arbitrary element of the iterated wreath product of cyclic groups C p i , p i ∈ N equals to 1. The commutator width of direct limit of wreath product of cyclic groups is found. Upper bounds for the commutator width ( c w ( G ) ) of a wreath product of groups are presented in this paper. A presentation in form of wreath recursion of Sylow 2-subgroups S y l 2 ( A 2 k ) of A 2 k is introduced. As a result, a short proof that the commutator width is equal to 1 for Sylow 2-subgroups of alternating group A 2 k , where k > 2 , the permutation group S 2 k , as well as Sylow p-subgroups of S y l 2 A p k as well as S y l 2 S p k ) are equal to 1 was obtained. A commutator width of permutational wreath product B ≀ C n is investigated. An upper bound of the commutator width of permutational wreath product B ≀ C n for an arbitrary group B is found. The size of a minimal generating set for the commutator subgroup of Sylow 2-subgroup of the alternating group is found. The proofs were assisted by the computer algebra system GAP.