A characterisation of A-simple groups

TL;DR

This paper characterizes finite groups with an elementary abelian r-group acting faithfully and transitively on simple components, providing insights into their structure and a new proof of McBride's Nonsolvable Signalizer Functor Theorem.

Contribution

It offers a new characterization of A-simple groups with elementary abelian r-group actions, advancing understanding of their fixed point subgroups and structure.

Findings

Characterization of G and fixed point subgroups under A-action

New proof of McBride's Nonsolvable Signalizer Functor Theorem

Insights into the structure of A-simple groups

Abstract

Let $A$ be an elementary abelian $r$-group with rank at least $3$ that acts faithfully on the finite $r'$-group $G$. Assume that $G$ is $A$-simple, so that $G = K_{1} \times\cdots\times K_{n}$ where $K_{1},\ldots,K_{n}$ is a collection of simple subgroups of $G$ that is permuted transitively by $A$. The purpose of this paper is to characterize $G$ and the collection of fixed point subgroups $\{ C_{G}(a) \;|\; a \in A^{\#} \}$. An application of this result will be a new proof of McBride's Nonsolvable Signalizer Functor Theorem.