Isomorphism testing for \(p\)-groups
type: Item
instance of: scholarly article
title: Isomorphism testing for \(p\)-groups
zbMATH Open document ID: 0824.20020
author: E. A. O'Brien
published in: Journal of Symbolic Computation
publication date: 1995-10-30T00:00:00Z
review text: Usually proving the isomorphism of two explicitly given groups is a difficult problem. Often it is easier to prove that two groups are not isomorphic by finding different group invariants. The paper solves this problem for \(p\)-groups. The standard \(p\)-group generation algorithm [the author, J. Symb. Comput. 9, 677-698 (1990; Zbl 0736.20001)] constructs specific extensions of \(p\)-groups, called immediate descendants. The construction requires knowledge of the generators of the automorphism group of the \(p\)-group. The \(p\)-quotient algorithm [\textit{G. Havas} and \textit{M. F. Newman}, Lect. Notes Math. 806, 211-230 (1980; Zbl 0432.20033)] uses the lower exponent- \(p\) central series (a variation of the lower central series) to construct a power commutator presentation of a given \(p\)-group. However, the resulting presentation is by no means canonical. Two different presentations may easily define isomorphic groups. In this paper both techniques are combined. A standard presentation of a given \(p\)-group is obtained by constructing a presentation with the \(p\)-group generation algorithm along its lower exponent-\(p\) central series. This presentation is canonical and depends only on the isomorphism type of the group. Isomorphism testing can now simply be done by comparing presentations. As a side effect generators of the automorphism group of the given \(p\)-group are constructed. Explicit construction of an isomorphism between the constructed and given presentation, and therefore the tested groups, is possible. However, the construction of the standard presentation is significantly slower than computing an arbitrary power commutator presentation with the \(p\)-quotient algorithm. Thus it should only be used for isomorphism testing, but not to simply obtain a finite presentation. This and other issues concerning performance and implementation are also discussed in the paper.
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20-04
Mathematics Subject Classification ID: 68W10
zbMATH DE Number: 638321
zbMATH Keywords: generators of automorphism groups
zbMATH Keywords: construction of isomorphisms
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: standard \(p\)-group generation algorithm
zbMATH Keywords: extensions of \(p\)-groups
zbMATH Keywords: immediate descendants
zbMATH Keywords: \(p\)-quotient algorithm
zbMATH Keywords: lower exponent-\(p\) central series
zbMATH Keywords: lower central series
zbMATH Keywords: power commutator presentations
zbMATH Keywords: standard presentations
zbMATH Keywords: isomorphism testing
zbMATH Keywords: performance
zbMATH Keywords: implementation
MaRDI profile type: Publication
full work available at URL: jsco.1994.1007
OpenAlex ID: W1989359050
DOI: 10.1006/JSCO.1994.1007
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description: scientific article; zbMATH DE number 638321