Asymptotic Results for Primitive Permutation Groups and Irreducible Linear Groups (original) (raw)

Bounding the composition length of primitive permutation groups and completely reducible linear groups

Journal of the London Mathematical Society

We obtain upper bounds on the composition length of a finite permutation group in terms of the degree and the number of orbits, and analogous bounds for primitive, quasiprimitive and semiprimitive groups. Similarly, we obtain upper bounds on the composition length of a finite completely reducible linear group in terms of some of its parameters. In almost all cases we show that the bounds are sharp, and describe the extremal examples.

On the maximum orders of elements of finite almost simple groups and primitive permutation groups

2013

We determine upper bounds for the maximum order of an element of a finite almost simple group with socle T in terms of the minimum index m(T) of a maximal subgroup of T: for T not an alternating group we prove that, with finitely many exceptions, the maximum element order is at most m(T). Moreover, apart from an explicit list of groups, the bound can be reduced to m(T)/4. These results are applied to determine all primitive permutation groups on a set of size n that contain permutations of order greater than or equal to n/4.

On the ranks of finite simple groups

Proceedings of the Third Biennial International Group Theory Conference (3BIGTC) Ferdowsi University of Mashhad, 2015

E E t t I t I t t t t t I t t t t rl t rl t rt t t t rl t t E I l Abstract Let Gbe a furite goup and X be a conjugacy class ofG, The rank of X in G, denotedby ran,k(G:X) is defined to be the minimal number of elements of X generating G. In this paper we review the basic results on generation of finite simple groups and we survey the recent developments on computing the ranks of l.urite simple groups.

Some Infinite Permutation Groups and Related Finite Linear Groups

Journal of the Australian Mathematical Society, 2016

This article began as a study of the structure of infinite permutation groups GGG in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups GGG are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup MMM which is a divisible abelian ppp-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a ppp-adic vector space associated with MMM. This leads to our second variation, which is a study of the finite linear groups that can arise.