On Simply Transitive Primitive Permutation Groups

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.

Generation and random generation: From simple groups to maximal subgroups

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 IBIS groups

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.

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.

On solvable factors of almost simple groups

Invariant theory for wreath product groups

We show that a finite permutation group containing a regular abelian self-normalizing subgroup is soluble.

On the tensor product of permutation modules

In this paper, we shall deal with periodic groups, in which each element has a prime power order. A group [Formula: see text] will be called a [Formula: see text]-group if each element of [Formula: see text] has a prime power order and for each [Formula: see text] there exists a positive integer [Formula: see text] such that each [Formula: see text]-element of [Formula: see text] is of order [Formula: see text]. A group [Formula: see text] will be called a [Formula: see text]-group if each element of [Formula: see text] has a prime power order and for each [Formula: see text] there exists a positive integer [Formula: see text] such that each finite [Formula: see text]-subgroup of [Formula: see text] is of order [Formula: see text]. Here, [Formula: see text] denotes the set of all primes dividing the order of some element of [Formula: see text]. Our main results are the following four theorems. Theorem 1: Let [Formula: see text] be a finitely generated [Formula: see text]-group. Then [Formula: see text] has only a finite number of normal subgroups of finite index. Theorem 4: Let [Formula: see text] be a locally graded [Formula: see text]-group. Then [Formula: see text] is a locally finite group. Theorem 7: Let [Formula: see text] be a locally graded [Formula: see text]-group. Then [Formula: see text] is a finite group. Theorem 8: Let [Formula: see text] be a [Formula: see text]-group satisfying [Formula: see text]. Then [Formula: see text] is a locally finite group.

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.

We study the minimal non-trivial subdegrees of finite primitive permutation groups that admit an embedding into a wreath product in product action, giving a connection with the same quantity for the primitive component. We discover that the primitive groups of twisted wreath type exhibit different (but interesting) behaviour from the other primitive types. 2000 Mathematics Subject Classification: 20B15

We show that, for any positive integer k, there are only finitely many finite groups, up to isomorphism, with exactly k conjugacy classes of elements of prime power order. This generalizes a result of E. Landau from 1903. The proof of our generalization makes use of the classification of finite simple groups.