\(RC_G(R)\)-signalizers in finite groups.
The author considers, for a prime \(r\), a finite \(r'\)-group \(G\) on which an \(r\)-group \(R\) acts as a group of automorphisms. The \(RC_G(R)\)-invariant subgroups of \(G\), called \(RC_G(R)\)-signalizers are studied and proved to have a serious influence on the structure of \(G\). This is a nicely written and rather concentrated paper which establishes a series of results (most of them too technical to present here) which are analogues, in this co-prime action setting, of older results related to \(p\)-local structure. In this reviewer's opinion, Theorem G is a most elegant result about the behavior of the generalized Fitting subgroup \(F^*(G)\): ``Suppose the noncyclic Abelian \(r\)-group \(R\) acts on the \(r'\)-group \(G\), let \(p>3\) be a prime and suppose that \([O^p(F^*(C_G(\alpha))),R]=1\) for all nontrivial elements \(\alpha\in R\). Then \([O^p(F^*(G)),R]=1\).'' The above leads to information on the structure of co-prime automorphism groups acting on simple non-Abelian groups (given that certain technical constraints are satisfied).
scientific article; zbMATH DE number 5696932