Finite groups with regular orbits on vector spaces
type: Item
instance of: scholarly article
title: Finite groups with regular orbits on vector spaces
zbMATH Open document ID: 0604.20014
DOI: 10.1016/0021-8693(86)90180-8
author: Peter Fleischmann
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let \(\pi\) '(A) be the set of all primes which not divide the order of the finite group A, \(p\in \pi'(A)\), \(V_ p(A):=\) all \(F_ qA\)-modules for arbitrary p-power q. \(S_ p(A):=\{V\in V_ p(A)|\) A has a regular orbit on \(V\}\). \(\tau(A)= \{p\in \pi '(A)|\) \(V_ p(A)=S_ p(A)\}\), \(\sigma (A)=\cap_{{\mathcal U}\leq A}\tau ({\mathcal U}).\) Theorem 1. If P is a p-group, \(p>2\), then \(\pi\) '(P)-\(\sigma\) (P)\(\subseteq \{2\}\). Theorem 2. Let \(\hat M\), \(\hat F\) denote Mersenne resp. Fermat primes. If P is a 2-group then \(\pi\) '(P)-\(\sigma\) (P)\(\subseteq \hat M\cup \hat F\). It is given some result on the connection between \(V_ p(A)\), \(S_ p(A)\), \(p\in \pi '(A)\).
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 3975187
zbMATH Keywords: character degrees
zbMATH Keywords: nilpotent groups
zbMATH Keywords: regular orbit
zbMATH Keywords: p-group
reviewed by: Yakov Berkovich
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90180-8
OpenAlex ID: W1966134857
cites work: Endliche Gruppen I
cites work: Endliche Gruppen mit vielen Untergruppen.
cites work: Auflösbare Gruppen auf denen nicht auflösbare Gruppen operieren
cites work: The existence of regular orbits for nilpotent groups
description: scientific article; zbMATH DE number 3975187