Michael J. J. Barry

Simple groups contain minimal simple groups

It is a consequence of the classification of finite simple groups that every non-abelian simple group contains a subgroup which is a minimal simple group.

Character degrees of simple groups

Gluck [2] and others have investigated the relationship between p(G), the set of primes dividing the degrees of the irreducible complex characters of G, and o(G), the greatest number of primes dividing the degree of a single irreducible character of G. when G is a finite solvable group. For such groups Gluck [2] has shown that 1 p(G)/ ,< (g(G))‘+ 100(G). In this paper we examine the relationship between these two quantities for G a finite nonabelian simple group. In order to state our results we need to introduce some notation. If G is a finite group and S is a subset of Irr(Gj, the set of irreducible characters of G, we will say S is a covering set of G if for every prime divisor p of / G 1 there is a character x in S such that p divides x( 1). If G has a covering set we will define the covering number of G, which we will denote by en(G), as the least number of elements in a covering set of G. Work of Michler [6, Theorem 3.31 has as a consequence that if G is a nonabelian simple group then G has a covering set; in other words, p(G) = n(G), where n(G) is the set of prime divisors of the order of G. For such groups then en(G) ,( 1 r(G)1 However, more is true.

On a Question of Glasby, Praeger, and Xia

A Jordan partition λ(m, n, p) = (λ1, λ2, … , λ m ) is a partition of mn associated with the expression of a tensor V m ⊗ V n of indecomposable KG-modules into a sum of indecomposables, where K is a field of characteristic p and G a cyclic group of p-power order. It is standard if λ i = m + n − 2i + 1 for all i. We answer a recent question of Glasby, Praeger, and Xia who asked for necessary and sufficient conditions for λ(m, n, p) to be standard.

Generators for Decompositions of Tensor Products of Modules associated with standard Jordan partitions

ABSTRACT If K is a field of finite characteristic p, G is a cyclic group of order q = pα, U and W are indecomposable KG-modules with dim U = m and dim W = n, and λ(m,n,p) is a standard Jordan partition of mn, we describe how to find a generator for each of the indecomposable components of the KG-module U ⊗ W.

Two-transitive actions on conjugacy classes

Every group acts transitively by conjugation on each of its conjugacy classes of elements. It is natural to wonder when this action becomes multiply transitive. In this paper, we determine all finite groups which act faithfully and 2-transitively on a conjugacy class of elements. We also give some consequences including a solvability criterion based on what fraction of elements belong to conjugacy classes upon which the group acts faithfully and 2–transitively.