Publication Date:
2016-04-06
Description:
Let $R$ be a group of prime order $r$ that acts on the $r'$ -group $G$ , let $RG$ be the semidirect product of $G$ with $R$ , let ${\mathbb {F}}$ be a field and $V$ be a faithful completely reducible $\mathbb {F}[{RG}]$ -module. Trivially, $C_{G}({R})$ acts on $C_{V}({R})$ . Let $K$ be the kernel of this action. What can be said about $K$ ? This question is considered when $G$ is soluble. It turns out that $K$ is subnormal in $G$ or $r$ is a Fermat or half-Fermat prime. In the latter cases, the subnormal closure of $K$ in $G$ is described. Several applications to the theory of automorphisms of soluble groups are given.
Print ISSN:
0024-6115
Electronic ISSN:
1460-244X
Topics:
Mathematics