Automorphisms of K-groups II

TL;DR

This paper extends the theory of automorphisms of finite K-groups with coprime automorphism groups, providing new proofs of key theorems in the field, including a special case of the Nonsolvable Signalizer Functor Theorem.

Contribution

It develops a general automorphism theory for finite K-groups and offers new proofs of existing theorems, enhancing understanding of group actions and signalizer functors.

Findings

Extended results of McBride on signalizer functors.

Developed a new proof of the Nonsolvable Signalizer Functor Theorem.

Provided a new proof for a special case of the Gorenstein-Lyons theorem.

Abstract

This work is a continuation of Automorphisms of $K$-groups I, P. Flavell, preprint. The main object of study is a finite $K$-group $G$ that admits an elementary abelian group $A$ acting coprimely. For certain group theoretic properties $\mathcal P$, we study the $AC_{G}(A)$-invariant $\mathcal P$-subgroups of $G$. A number of results of McBride, 'Near solvable signalizer functors on finite groups' J. Algebra {\bf 78}(1) (1982) 181-214 and 'Nonsolvable signalizer functors on finite groups', J. Algebra {\bf 78}(1) (1982) 215-238 are extended. One purpose of this work is to build a general theory of automorphisms, one of whose applications will be a new proof of the Nonsolvable Signalizer Functor Theorem. As an illustration, this work concludes with a new proof of a special case of that theorem due to Gorenstein and Lyons.