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Finite \(p\)-groups of maximal class with ‘large’ automorphism groups. (English) Zbl 1369.20019

The classification of the \(p\)-groups of maximal class is a wide open problem with a long history. A central tool in this investigation is the coclass graph \(\mathcal G\), whose vertices are given by finite \(p\)-groups of maximal class, one for each isomorphism type, and two vertices \(G\) and \(H\) are joined if and only if \(H/\gamma(H)\cong G,\) being \(\gamma(H)\) the last non-trivial term of the lower central series of \(H\). The coclass conjecture W suggests that \(\mathcal G\) can be determined from a finite subgraph using certain periodic patterns [B. Eick et al., Int. J. Algebra Comput. 23, No. 5, 1243–1288 (2013; Zbl 1298.20020)]. In this paper, the authors consider the subgraph \(\mathcal G^*\) of \(\mathcal G\) associated with those \(p\)-groups of maximal class whose automorphism group orders are divisible by \(p - 1\). They describe the broad structure of \(\mathcal G^*\) by determining its so-called skeleton. They investigate the smallest interesting case \(p = 7\) in more detail using computational tools and propose an explicit version of conjecture W for \(\mathcal G^*\) for arbitrary \(p \geq 7.\) As remarked by the authors, there is only very little evidence for conjecture W so far and all the available evidence is in coclass trees of finite width. The results in this paper are the first explicit evidence in support of conjecture W for a coclass graph of infinite width.

MSC:

20D15 Finite nilpotent groups, \(p\)-groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20F28 Automorphism groups of groups
20E18 Limits, profinite groups
20F05 Generators, relations, and presentations of groups

Citations:

Zbl 1298.20020

Software:

GAP; Coclass; SymbCompCC
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Full Text: DOI

References:

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