Wreath products and enlargements of groups
type: Item
instance of: scholarly article
title: Wreath products and enlargements of groups
zbMATH Open document ID: 0787.20019
DOI: 10.1016/0012-365X(93)90560-G
published in: Discrete Mathematics
publication date: 1994-05-24T00:00:00Z
review text: Let \(N^*_ G(\Delta) = \bigcap_{g\in G} g^{-1}\Delta g\) the autonormalizer of \(\Delta\) in a group \(G\). The subgroup \(\Delta\) is coreless in \(G\) if \(N^*_ G(\Delta) = 1\). The group \(G\) called an enlargement of a \(\theta\) group \(H\) by a group \(J\) if there exist an exact sequence \(1 \to H {\overset\lambda\longrightarrow} G \to J \to 1\) and a normal subgroup \(\Delta\) of \(\lambda(H)\) such that \(\lambda(H)/\Delta \approx \Theta\) and is coreless in \(G\). In this paper the author investigates the relationship between group enlargements and wreath products. By the Extension Lemma of Suzuki (Let \(H\) and \(J\) be any two groups such that \(H\) is centreless. Then every outer action of \(J\) on \(H\) is uniquely realized by an extension of \(H\) by \(J\)) and the Schreier Conjecture (Every finite simple group has a solvable outer automorphism group) he gets the following consequences: The direct product \(H\times J\) is the only extension of a nonabelian simple group \(H\) by a nonabelian simple group \(J\).
reviewed by: Piroska Lakatos
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20D05
zbMATH DE Number: 429130
zbMATH Keywords: autonormalizer
zbMATH Keywords: group enlargements
zbMATH Keywords: wreath products
zbMATH Keywords: Extension Lemma of Suzuki
zbMATH Keywords: Schreier Conjecture
zbMATH Keywords: finite simple group
zbMATH Keywords: outer automorphism group
MaRDI profile type: Publication
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author: Shreeram S. Abhyankar
description: scientific article; zbMATH DE number 429130