Automorphisms and stem extensions
type: Item
instance of: scholarly article
title: Automorphisms and stem extensions
zbMATH Open document ID: 0555.20021
DOI: 10.1016/0021-8693(84)90093-0
author: Michael D. Fry
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: A stem extension of a finite group G is a short exact sequence \(e: O\to A\to E\to G\to O\) of finite groups such that \(A\leq Z(E)\cap E'\). A stem cover (or representation group) of G is any stem extension \(e: O\to A\to E\to G\to O\) of G such that \(| A| =| H^ 2(G,{\mathbb{C}}^ x)|\). A finite group G has property \({\mathcal L}\) if for every \(\alpha\in aut(G)\), there is a stem cover \(e: O\to A\to E\to G\to O\) and an automorphism \(\beta\in aut(E)\) that ''induces'' \(\alpha\) (i.e. \(\beta\) leaves A invariant). Section 1 of this paper studies the natural action of aut(G) on the stem extensions of G and deduces connections between this aut(G)-set and the property \({\mathcal L}\). More machinery is developed in Sections 2 and 3. These results are used in Sections 4-5 to obtain deeper results concerning the action of aut(G) on the stem extensions of G. An example of such a result is Corollary 4.5: If every Sylow subgroup of G has property \({\mathcal L}\), then G also has property \({\mathcal L}.\) Sections 6-7 of this paper are devoted to proving Theorem 7.1: Suppose that G is elementary abelian of order \(2^ r\). Then G has property \({\mathcal L}\) if and only if \(r\leq 2\).
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 3884374
zbMATH Keywords: stem extension
zbMATH Keywords: short exact sequence
zbMATH Keywords: stem cover
zbMATH Keywords: action
zbMATH Keywords: Sylow subgroup
reviewed by: Morton E. Harris
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90093-0
OpenAlex ID: W2091262563
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description: scientific article; zbMATH DE number 3884374