On the structure of a group whose orbits on a finite module are the orbits of a proper subgroup
type: Item
instance of: scholarly article
title: On the structure of a group whose orbits on a finite module are the orbits of a proper subgroup
zbMATH Open document ID: 0601.20011
DOI: 10.1016/0021-8693(85)90190-5
author: Trevor Hawkes
author: Graham Jones
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: The authors prove a result they call the Orbit Theorem, which has been used by \textit{T. Hawkes} and \textit{J. Humphreys} [J. Algebra 94, 382-387 (1985; Zbl 0594.20007)] to give a criterion for existence of normal complements. Let p, q be distinct primes, let G be a finite solvable group and let Q be a normal elementary abelian subgroup of G, complemented in G by a subgroup H. Let V be a kG-module, faithful for Q, in which the G-orbit of each element coincides with its H-orbit. The Orbit Theorem asserts that under these conditions, \([Q,O_{q'}(H)]=1\).
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D40
zbMATH DE Number: 3968870
zbMATH Keywords: modular representations
zbMATH Keywords: orbits
zbMATH Keywords: existence of normal complements
zbMATH Keywords: finite solvable group
zbMATH Keywords: kG-module
zbMATH Keywords: Orbit Theorem
reviewed by: William H. Gustafson
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90190-5
OpenAlex ID: W1979948536
cites work: Q5577154
cites work: A character-theoretic criterion for the existence of normal complements to subgroups of finite groups
cites work: Endliche Gruppen I
cites work: The Influence of Nilpotent Subgroups on the Nilpotent Length and Derived Length of a Finite Group
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description: scientific article; zbMATH DE number 3968870