Realizing finite groups as automizers
type: Item
instance of: scholarly article
title: Realizing finite groups as automizers
zbMATH Open document ID: 1543.20018
author: Sylvia Bayard
author: Justin Lynd
published in: Journal of Group Theory
publication date: 2024-07-02T00:00:00Z
review text: The authors study the realization of finite groups by automizers of subgroups of finite groups. That is, given a finite group \(A\), they study when it is possible to find a finite group \(G\) and a subgroup \(U \leq G\) such that \(A \cong {\Aut}_G(U) \cong N_G(U)/C_G(U)\). The answer to this question is \lq always possible\rq \ for trivial reasons: choose a faithful action of \(A\) on an elementary abelian \(p\)-group \(U\) and take for \(G\) the semidirect product of \(U\) by \(A\). In this case, \(U\) is normal in \(G\). The main result shows that it is possible to realize \(A\) as \({\Aut}_G(U)\), where \(U\) is very far from being normal. It is shown that any finite group \(A\) is realizable as the automizer in a finite perfect group \(G\) of an abelian subgroup whose conjugates generate \(G\). The construction uses techniques from fusion systems on arbitrary finite groups, most notably certain realization results for fusion systems of the type studied originally by \textit{S. Park} [Arch. Math. 94, No. 5, 405--410 (2010; Zbl 1243.20025); Proc. Am. Math. Soc. 144, No. 8, 3291--3294 (2016; Zbl 1367.20016)].
reviewed by: Andrea Lucchini
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D45
zbMATH DE Number: 7874816
zbMATH Keywords: automizers
zbMATH Keywords: perfect groups
zbMATH Keywords: fusion systems
MaRDI profile type: Publication
cites work: Q3114249
cites work: Q3077959
cites work: Q3913574
cites work: Realizing a fusion system by a single finite group.
cites work: Realizing fusion systems inside finite groups
cites work: Fusion systems and constructing free actions on products of spheres
DOI: 10.1515/JGTH-2022-0145
description: scientific article; zbMATH DE number 7874816