Modular Frobenius groups
type: Item
instance of: scholarly article
title: Modular Frobenius groups
zbMATH Open document ID: 0868.20008
DOI: 10.1007/BF02568315
author: Eric B. Kuisch
author: Robert W. van der Waall
published in: Manuscripta Mathematica
publication date: 1997-06-16T00:00:00Z
full work available at URL: https://eudml.org/doc/156202
review text: Let \(G\) be a finite group and let \(N\) be a proper non-trivial normal subgroup of \(G\). Furthermore, let \(K\) be a splitting field of characteristic \(p\) for \(N\). Then \(G\) is called a (\(p\)-)modular Frobenius group if for every non-trivial irreducible \(KN\)-module \(V\) the induced module \(V^G\) is irreducible. Note that modular Frobenius groups in characteristic zero are just Frobenius groups. In particular, the Frobenius kernel \(N\) is nilpotent, by a result of Thompson. In the paper under review modular Frobenius groups are studied for a prime \(p\). As a main result the authors prove that \(N\) is solvable if \(p\) is odd while the non-cyclic composition factors of \(N\) are isomorphic to \(\text{PSL}(2,3^{2^t})\), \(t\geq 1\), if \(p=2\). The proof depends on the classification of finite simple groups. For any \(t\geq 1\) examples are given in which the kernel \(N\) involves \(\text{PSL}(2,3^{2^t})\). The authors also investigate the structure of solvable modular Frobenius groups.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 980704
zbMATH Keywords: \(\text{PSL}(2,3^ t)\)
zbMATH Keywords: Frobenius kernels
zbMATH Keywords: non-cyclic composition factors
zbMATH Keywords: solvable modular Frobenius groups
reviewed by: Wolfgang Willems
MaRDI profile type: Publication
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OpenAlex ID: W2001990220
description: scientific article; zbMATH DE number 980704