A computer program for the calculation of a covering group of a finite group
type: Item
instance of: scholarly article
title: A computer program for the calculation of a covering group of a finite group
zbMATH Open document ID: 0552.20006
DOI: 10.1016/0022-4049(85)90046-5
published in: Journal of Pure and Applied Algebra
publication date: 1985-01-01T00:00:00Z
review text: This is a direct continuation of the same author's paper [Computational group theory, Proc. Symp., Durham/Engl. 1982, 307-319 (1984; Zbl 0544.20004)]. In the calculation of the multiplier M(P) of a p-Sylow subgroup P of a permutation group G by the NQ-algorithm actually a certain covering group \(\hat P\) of P is obtained. By methods described in the above mentioned paper, a subgroup X of M(P) can be constructed such that the p-part \(M=M(G)_ p\) of the multiplier of G is obtained as M(P)/X. Forming \(\hat P/\)X one gets a presentation of a certain extension D of M by P. D corresponds to an element \([D]\in H^ 2(P,M)\). Its image under the corestriction map \(Cor_{P,G}: H^ 2(P,M)\to H^ 2(G,M)\) is found, defining an extension \(\hat G_ p\) of M by G. Finally defining relations of a covering group \(\hat G\) of G are composed from those of the \(\hat G_ p\), and, if there are several covering groups these are constructed from the first one. The theory involved in particular in the actual construction of \(Cor_{P,G}\) is given in detail as well as a short characterization of the new programs that had to be implemented in addition to those already needed in the previous paper. The paper closes with some examples of the use of the implementation, giving presentations of the (unique) covering groups of PSL(3,4), \(M_{22}\), and PSL(4,3). (Reviewer's remark: The author has meanwhile continued this very interesting line of research by extending his methods to the calculation of cohomology groups for non-trivial modules and corresponding extensions. This will be reported about in a forthcoming paper: The mechanical computation of first and second cohomology groups.)
Mathematics Subject Classification ID: 20C25
Mathematics Subject Classification ID: 20-04
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 3879128
zbMATH Keywords: multiplier
zbMATH Keywords: p-Sylow subgroup
zbMATH Keywords: permutation group
zbMATH Keywords: NQ-algorithm
zbMATH Keywords: presentation
zbMATH Keywords: corestriction
zbMATH Keywords: relations
zbMATH Keywords: covering group
zbMATH Keywords: cohomology groups
author: Derek F. Holt
reviewed by: Joachim Neubüser
MaRDI profile type: Publication
full work available at URL: 0022-4049(85)90046-5
OpenAlex ID: W2014270052
cites work: Q3240881
cites work: Q3333222
cites work: On an Algorithm for Finding a Base and a Strong Generating Set for a Group Given by Generating Permutations
cites work: Q3950713
description: scientific article; zbMATH DE number 3879128