Computing the order of a solvable permutation group
type: Item
instance of: scholarly article
title: Computing the order of a solvable permutation group
zbMATH Open document ID: 0701.20001
DOI: 10.1016/S0747-7171(08)80083-1
author: Charles C. Sims
published in: Journal of Symbolic Computation
publication date: 1990-01-01T00:00:00Z
review text: If a group G of permutations is given by a set of generators usually the first step for its investigation is the determination of a base and a strong generating system by one of the versions of the Schreier-Sims method [cf. \textit{J. S. Leon}; Math. Comput. 35, 941-974 (1980; Zbl 0444.20001)]. In this article the author describes a new method, which works if G is soluble (it runs into an endless cycle if G is not soluble). The method is not only considerably faster in many cases tested, but it also constructs at the same time a ``polycyclic generating sequence'', i.e. a sequence of generators \(y_ 1,...,y_ t\) of G such that with \(H_ i=<y_ i,...,y_ t>\) one has \(H_ i\triangleleft H_{i-1}\) for \(1<i\leq t\) and \(H_ 1=G\). There are a number of computational problems, e.g. the determination of the conjugacy classes, for which rather efficient solutions are known for soluble groups using such a polycyclic generating sequence, while they are notoriously difficult using only base and strong generating set. Like other computational methods for soluble groups this one exploits the fact that an orbit of a normal subgroup of a group G is a block for G.
Mathematics Subject Classification ID: 20B40
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 20-04
Mathematics Subject Classification ID: 20B05
zbMATH DE Number: 4148389
zbMATH Keywords: permutations
zbMATH Keywords: generators
zbMATH Keywords: strong generating system
zbMATH Keywords: Schreier-Sims method
zbMATH Keywords: conjugacy classes
zbMATH Keywords: soluble groups
zbMATH Keywords: polycyclic generating sequence
reviewed by: Joachim Neubüser
MaRDI profile type: Publication
cites work: On the length of subgroup chains in the symmetric group
cites work: Q3792860
cites work: The solvable length of a solvable linear group
cites work: A compact representation for permutation groups
cites work: On an Algorithm for Finding a Base and a Strong Generating Set for a Group Given by Generating Permutations
full work available at URL: s0747-7171(08)80083-1
OpenAlex ID: W2045451947
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description: scientific article; zbMATH DE number 4148389