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Investigating \(p\)-groups by coclass with GAP. (English) Zbl 1167.20011

Kappe, Luise-Charlotte (ed.) et al., Computational group theory and the theory of groups. AMS special session on computational group theory, Davidson, NC, USA, March 3–4, 2007. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4365-9/pbk). Contemporary Mathematics 470, 45-61 (2008).
There is no hope of classifying finite \(p\)-groups by increasing nilpotence class, as the collection of all finite \(p\)-groups of class 2 is already as complicated, in a precise technical sense, as that of all finite groups. C. R. Leedham-Green and M. F. Newman [Arch. Math. 35, 193-202 (1980; Zbl 0437.20016)] suggested to look at these groups by coclass instead. Here the coclass of a group of order \(p^n\) and class \(c\) is \(n-c\). The coclass conjectures they formulated were later proved by C. R. Leedham-Green [J. Lond. Math. Soc., II. Ser. 50, No. 1, 43-48 (1994; Zbl 0817.20031)] and A. Shalev [Invent. Math. 115, No. 2, 315-345 (1994; Zbl 0795.20009)], in a milestone in the theory of finite \(p\)-groups.
The paper under review is concerned with the next bold step. This is a conjecture that says that, given a prime \(p\) and a number \(r\), one can divide finite \(p\)-groups of coclass \(r\) into finitely many families. The groups in each family can be given a common, parametrised presentation. Moreover, many important structural invariants of the groups in a family (such as the automorphism group or the Schur multiplicator) can also be described in a parametrised way. Although work on this conjecture, which is actually aiming at a classification of finite \(p\)-groups by coclass, is still at the beginning, even partial results in this direction would yield powerful tools for studying a variety of problems in the theory of finite \(p\)-groups.
The paper describes an algorithm, which has been implemented in GAP, which allows to investigate the conjecture, and describes several applications of it. We refer to this very well laid out paper for the very interesting details.
For the entire collection see [Zbl 1147.20002].

MSC:

20D15 Finite nilpotent groups, \(p\)-groups
20-04 Software, source code, etc. for problems pertaining to group theory
20D45 Automorphisms of abstract finite groups
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