Normalizers of \(p\)-subgroups in finite groups.
type: Item
instance of: scholarly article
title: Normalizers of \(p\)-subgroups in finite groups.
zbMATH Open document ID: 1176.20012
author: Chris Bates
author: Peter J. Rowley
published in: Archiv der Mathematik
publication date: 2009-05-06T00:00:00Z
review text: If \(G\) is a permutation group stored in a computer as a set of generating permutations, and \(H\) is a subgroup of \(G\), the standard way to compute \(N_G(H)\) is to compute a base and strong generating set first. This permits use of the powerful methods pioneered by Sims in the late 1960s to compute almost anything in the group. These methods are so efficient that if \(G\) is a matrix group, the usual way to proceed is to first convert \(G\) into a permutation group. However, it often turns out that the resulting permutations are unmanageably huge. Therefore it is desirable to develop other methods for matrix groups (or more generally, black-box groups). The paper under review reports on an implementation of such methods for \(H\) a \(p\)-group. The method for \(p=2\) is part of the folklore [see for example \textit{G. Butler}, J. Algebra 69, 67-81 (1981; Zbl 0457.20024)], but is here presented more systematically, and completely automated. Essentially, a number of composition series for \(H\) are computed, and for each one a set of random elements are found which centralize each composition factor. Since each composition factor is \(C_2\), this is essentially an involution centralizer calculation. For \(p\) odd, the same strategy is employed, but using the authors' method [described in Arch. Math. 85, No. 6, 485-489 (2005; Zbl 1137.20301)] for finding centralizers of odd-order elements.
Mathematics Subject Classification ID: 20C40
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20D06
zbMATH DE Number: 5551217
zbMATH Keywords: normalizers
zbMATH Keywords: centralizers
zbMATH Keywords: \(p\)-subgroups
zbMATH Keywords: computational methods in group theory
zbMATH Keywords: matrix groups
zbMATH Keywords: black-box groups
zbMATH Keywords: composition series
reviewed by: Robert A. Wilson
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: s00013-008-2737-2
OpenAlex ID: W2112896218
DOI: 10.1007/S00013-008-2737-2
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description: scientific article; zbMATH DE number 5551217