## Abstract

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 픽 be a field and*V*be a faithful completely reducible 픽[*RG*]-module. Trivially,*C*(_{G}*R*) acts on*C*(R). Let_{V}*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.Original language | English |
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Pages (from-to) | 623-650 |

Number of pages | 28 |

Journal | London Mathematical Society. Proceedings |

Volume | 112 |

Issue number | 4 |

DOIs | |

Publication status | Published - 5 Apr 2016 |

## Keywords

- Automophisms
- soluble group

## ASJC Scopus subject areas

- Algebra and Number Theory