Saturated formations and products of connected subgroups.
type: Item
instance of: scholarly article
title: Saturated formations and products of connected subgroups.
zbMATH Open document ID: 1243.20020
published in: Journal of Algebra
publication date: 2011-12-08T00:00:00Z
review text: If \(\mathcal C\) is a non-empty class of groups, two subgroups \(A\) and \(B\) of a group \(G\) are said to be \(\mathcal C\)-connected if \(\langle a,b\rangle\in\mathcal C\) for all \(a\in A\) and \(b\in B\). If \(\pi\) and \(\rho\) are sets of primes, then \(S_\pi\) (\(S_\rho\)) denotes the class of finite soluble \(\pi\)-groups (\(\rho\)-groups) and \(S_\pi S_\rho\) the class of extensions of groups in \(S_\pi\) by groups in \(S_\rho\). If \(A\) and \(B\) are soluble subgroups of a finite group \(G\) such that \(G=AB\), then \(A\) and \(B\) are \(S_\pi S_\rho\)-connected if and only if \(O^\rho(B)\) centralises \(AO_\pi(G)/O_\pi(G)\), \(O^\rho(A)\) centralises \(BO_\pi(G)/O_\pi(G)\) and \(G\in S_{\pi\cup\rho}\) (Theorem 1). -- This result is then extended to so-called nilpotent-like Fitting formations of soluble groups (Theorem 2). These are non-empty saturated formations \(\mathcal F\) locally defined by a formation function \(f\) given by \(f(p)=S_{\pi(p)}\) for a set of primes \(\pi(p)\) containing \(p\), for each prime \(p\) belonging to the characteristic of \(\mathcal F\). Similar results hold for finite groups that are products of arbitrarily many pairwise permutable \(\mathcal F\)-connected \(\mathcal F\)-subgroups.
reviewed by: Hans Lausch
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 5987858
zbMATH Keywords: \(\mathcal C\)-connectedness
zbMATH Keywords: finite soluble groups
zbMATH Keywords: Fitting formations
zbMATH Keywords: saturated formations
zbMATH Keywords: products of subgroups
author: M. Pilar Gállego
author: Peter Hauck
author: M. D. Pérez-Ramos
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.09.012
OpenAlex ID: W2062766877
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DOI: 10.1016/J.JALGEBRA.2010.09.012
description: scientific article; zbMATH DE number 5987858