Existence, algorithms, and asymptotics of direct product decompositions. I.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1277.20022
publication date: 2012-08-28T00:00:00Z
full work available at URL: https://arxiv.org/abs/1005.0548
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20E22
zbMATH DE Number: 6072866
zbMATH Keywords: finite groups
zbMATH Keywords: direct products
zbMATH Keywords: direct decompositions
zbMATH Keywords: products of subgroups
zbMATH Keywords: group varieties
zbMATH Keywords: automorphism invariant subgroups
zbMATH Keywords: Remak decompositions
zbMATH Keywords: graded chief series
author: James B. Wilson
MaRDI profile type: Publication
OpenAlex ID: W3104785372
arXiv ID: 1005.0548
DOI: 10.1515/GCC-2012-0007
title: Existence, algorithms, and asymptotics of direct product decompositions. I.
published in: Groups, Complexity, Cryptology
review text: Let \(G\) be a finite group and fix a set \(\Omega\) of automorphisms of \(G\). We call \(H\) an \(\Omega\)-subgroup (respectively, \((\Omega\cup G)\)-subgroup) if \(H\) is a subgroup (respectively, normal subgroup) of \(G\) invariant under all \(\omega\in\Omega\). The object of this paper (and a paper to follow) is to present a constructive theory of direct decompositions of \(G\) into products of \(\Omega\)-subgroups, in particular, a decomposition into \(\Omega\)-indecomposables (called a Remak decomposition). A key idea is the concept of an \(\Omega\)-graded subgroup \(N\). This is a normal subgroup \(N\) such that if \(\mathcal H\) is the set of factors in a direct \(\Omega\)-decomposition of \(G\), then \(\{H\cap N\mid H\in\mathcal H\}\setminus\{1\}\) and \(\{HN/N\mid H\in\mathcal H\}\setminus\{N/N\}\) give direct (\(\Omega\cup G)\)-decompositions for \(N\) and \(G/N\), respectively.NEWLINENEWLINE Let \(\mathfrak X\) be a class of finite \(\Omega\)-groups closed under taking isomorphic images, direct \(\Omega\)-products and direct \(\Omega\)-factors. An up-\(\Omega\)-grader is an idempotent function \(G\mapsto\mathfrak X(G)\) of \(\mathfrak X\) into itself such that \(\mathfrak X(G)\) is \(\Omega\)-graded and for any direct \(\Omega\)-factor \(H\) of \(G\) we have \(\mathfrak X(H)=H\cap\mathfrak X(G)\). The idea is that starting from an \(\Omega\)-graded subgroup \(N\) we should be able to reconstruct a direct \(\Omega\)-decomposition of \(G\) from direct \((\Omega\cup G)\)-decompositions of \(N\) and \(G/N\).NEWLINENEWLINE Some typical theorems are the following. (Theorem 1): Every \(\Omega\)-group \(G\) has an \(\Omega\)-graded chief series. Thus either \(G\) is a characteristically simple \(\Omega\)-group and the Remak \(\Omega\)-decompositions are known (Theorem 6) or \(G\) has a proper nontrivial \(\Omega\)-graded subgroup \(N\). (Corollary to Theorem 2): an \(\Omega\)-group \(G\) is indecomposable if there is an up-\(\Omega \)-grader with \(\zeta_1(G)\le…
reviewed by: John D. Dixon
description: scientific article; zbMATH DE number 6072866