Isomorphism testing for \(p\)-groups
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.
standard \(p\)-group generation algorithm
scientific article; zbMATH DE number 638321