Computing automorphism groups of finite groups
type: Item
instance of: scholarly article
zbMATH Open document ID: 0995.20007
author: Derek F. Holt
publication date: 2002-06-30T00:00:00Z
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20B40
Mathematics Subject Classification ID: 20-04
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 1683593
zbMATH Keywords: automorphism groups
zbMATH Keywords: computation
zbMATH Keywords: finite groups
zbMATH Keywords: algorithms
zbMATH Keywords: chief factors
zbMATH Keywords: transitive permutation groups
MaRDI profile type: Publication
title: Computing automorphism groups of finite groups
review text: This paper gives an outline (a longer version is promised) of a new algorithm for computing the automorphism group of a finite group. Similar to the author's algorithm for subgroups [\textit{J. J. Cannon, B. C. Cox, D. F. Holt}, J. Symb. Comput. 31, No. 1-2, 149-161 (2001; Zbl 0984.20002)] it uses tabulated information to obtain the automorphism group for the factor of the largest normal solvable subgroup, and then lifts the result via steps over elementary Abelian chief factors to the whole group.NEWLINENEWLINENEWLINEThis lifting step generalizes the method for \(p\)-groups of \textit{E. A. O'Brien} [Computational algebra and number theory, Math. Appl., Dordr. 325, 83-90 (1995; Zbl 0836.20002)] and the method for solvable groups of \textit{M. J. Smith} [ Doctoral Thesis, Australian National University (1994)].NEWLINENEWLINENEWLINEThe main calculations concern the automorphism group of \(KG\)-modules, a potential bottleneck is to determine which group automorphisms induce particular module isomorphisms.NEWLINENEWLINENEWLINEThe paper closes with a list of performance figures that show that the method proposed is certainly practical for groups of size \(10^6\) (and likely larger groups as well). In fact with a few obvious exceptions (mainly \(p\)-groups) it succeeded on a list of all transitive groups of degree up to 30 determined by the reviewer [Konstruktion transitiver Permutationsgruppen, Aachener Beiträge zur Mathematik 18 (1996; Zbl 0955.20002)].NEWLINENEWLINEFor the entire collection see [Zbl 0959.00030].
reviewed by: A. Hulpke
description: scientific article; zbMATH DE number 1683593