On normal subgroups which are direct products
type: Item
instance of: scholarly article
title: On normal subgroups which are direct products
zbMATH Open document ID: 0594.20018
DOI: 10.1016/0021-8693(84)90203-5
author: Fletcher Gross
author: L. G. Kovács
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: Let \(G\) be a group and \(M\) be a normal subgroup of \(G\). In this paper the authors consider the extension problem when \(G/M\) is a known group and \(M\) is a direct product of nonabelian simple groups. Assume that \(M\) is a direct product of simple groups \(S_ i\) such that \(\{S_ i\}\) is a conjugacy class of subgroups if \(G\). If \(M\) is a minimal normal subgroup of a finite group \(G\) and if \(M\) is nonabelian, then we have this situation. Let \(M=S\times K\) where \(S\) is a simple group, and set \(N=N_ G(K)\). Then, the group \(G\) is completely determined by the groups \(G/M\) and \(N/K\) together with an obvious homomorphism from \(N/K\) into \(G/M\) (Theorem 4.1). The authors prove that the extension of \(G\) over \(M\) splits if the extension of \(N/K\) over \(M/K\) splits. There is a one-to-one correspondence between conjugacy classes of complements to \(M\) in \(G\) and conjugacy classes of complements to \(M/K\) in \(N/K\). More generally, there is a bijection between conjugacy classes in \(N/K\) of subgroups \(L/K\) such that \(N/K=(M/K)(L/K)\) and conjugacy classes in \(G\) of subgroups \(H\) such that \(G=HM\) and \(H\cap M=\prod (H\cap M)/(H\cap K_ i)\) (the complete direct product) where \(\{K_ i\}\) is the set of all conjugate subgroups of \(K\) (Cor. 4.4 of Theorem 4.2). The construction of \(G\) from \(\alpha: N/K\to G/M\) is given by what the authors call the induced extension. Let \(\alpha\) be a homomorphism of a group \(A\) into a group \(B\). Let \(C=\alpha (A)\), let \(I\) be the set of right cosets of \(C\) in \(B\), let \(\rho\) be the permutation representation of \(B\) on \(I\), let \(P=\rho(B)\), and let \(T\) be a fixed right transversal of \(C\) in \(B\). For each \(x\in B\), let \(x_ T\) be the function defined on \(I\) by \(x_ T(Ct)=u_ T(tx^{-1})xt^{-1}\) where \(t\in T\) and \(u_ T(y)=Cy\cap T\). Let \(\lambda_ T\) be the mapping from \(B\) into the wreath product \(C Wr(P,I)\) defined by \(\lambda_ T(x)=\rho(x)x_ T\). …
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20E22
zbMATH DE Number: 3955204
zbMATH Keywords: extension problem
zbMATH Keywords: direct products of nonabelian simple groups
zbMATH Keywords: conjugacy classes of subgroups
zbMATH Keywords: minimal normal subgroups
zbMATH Keywords: conjugacy classes of complements
zbMATH Keywords: induced extensions
zbMATH Keywords: wreath products
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90203-5
OpenAlex ID: W2038376074
cites work: On groups without Abelian composition factors
cites work: Complements of Normal Subgroups in Infinite Groups
cites work: Zur Erweiterungstheorie der endlichen Gruppen.
cites work: Cohomological topics in group theory.
cites work: Endliche Gruppen I
cites work: Twisted wreath products of groups
cites work: On the structure of standard wreath products of groups
cites work: Q3926778
description: scientific article; zbMATH DE number 3955204