Complete finite semidirect products and wreath products.
type: Item
instance of: scholarly article
title: Complete finite semidirect products and wreath products.
zbMATH Open document ID: 1223.20015
author: Ben Brewster
author: Elizabeth Wilcox
published in: Archiv der Mathematik
publication date: 2011-06-01T00:00:00Z
review text: A group \(G\) is `complete' if \(Z(G)=\{1\}\) and \(\Aut(G)=\mathrm{Inn}(G)\). This paper uses group cohomology to give a sufficient condition for a finite semidirect product \(G=N\rtimes H\) with \(C_G(N)\leq N\) to be complete and proves a partial converse. Using these results the authors fully characterize complete permutational wreath products proving the following result (Theorem 2.3): Let \(G\), \(H\) be non-trivial finite groups and let \(H\) be a permutation group acting faithfully on a finite set \(\Omega\). Let \(W=G\wr H\) and let \(B\) represent the base of \(W\). Then \(W\) is complete if and only if (a) \(G\) is complete; (b) the base \(B\) is characteristic in \(W\); (c) the image of \(H\) is self-normalizing in \(\mathrm{Out}(B)\).
reviewed by: Enrico Jabara
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20E22
zbMATH DE Number: 5902951
zbMATH Keywords: automorphisms
zbMATH Keywords: complete groups
zbMATH Keywords: semidirect products
zbMATH Keywords: wreath products
zbMATH Keywords: products of subgroups
MaRDI profile type: Publication
full work available at URL: s00013-011-0237-2
OpenAlex ID: W2095272728
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DOI: 10.1007/S00013-011-0237-2
description: scientific article; zbMATH DE number 5902951