Finite groups with large automizers for their nonabelian subgroups
type: Item
instance of: scholarly article
title: Finite groups with large automizers for their nonabelian subgroups
zbMATH Open document ID: 0903.20011
author: V. D. Mazurov
author: Marian Deaconescu
published in: Archiv der Mathematik
publication date: 1999-01-05T00:00:00Z
review text: For a subgroup \(H\) of a finite group \(G\) denote \(\Aut_G(H)=N_G(H)/C_G(H)\) to be the automizer of \(H\) in \(G\). Since \(\text{Inn}(H)\) is isomorphic to a subgroup of \(\Aut_G(H)\) which is embedded in \(\Aut(H)\), the authors identify \(\Aut_G(H)\) as small if \(\Aut_G(H)\cong\text{Inn}(H)\) and large if \(\Aut_G(H)\cong\Aut(H)\). H. Zassenhaus proved that a finite group \(G\) is abelian if and only if the automizers of all abelian subgroups are small. By replacing `small' with `large', this was supplemented by \textit{H. Bechtell}, \textit{G. Silberberg} and \textit{M. Deaconescu} [Can. Math. Bull. 40, No. 3, 266-270 (1997; Zbl 0891.20016)] and extended to all subgroups by \textit{J. Lennox} and \textit{J. Wiegold} [Rend. Semin. Mat. Univ. Padova 89, 83-86 (1993; Zbl 0802.20026)]. In this article, the authors prove that the collection of finite groups with large automizers for their nonabelian subgroups comprises those which are holomorphs of groups of odd prime order.
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 20E36
zbMATH DE Number: 1145409
zbMATH Keywords: automizers of Abelian subgroups
zbMATH Keywords: holomorphs of groups of prime orders
zbMATH Keywords: finite groups with large automizers
zbMATH Keywords: nonabelian subgroups
reviewed by: Homer F. Bechtell
MaRDI profile type: Publication
full work available at URL: s000130050143
OpenAlex ID: W1988828195
DOI: 10.1007/S000130050143
description: scientific article; zbMATH DE number 1145409