Finite p-groups of maximal class with 'large' automorphism groups

Heiko Dietrich, Bettina Eick

Research output: Contribution to journalArticleResearchpeer-review

Abstract

The classification of p-groups of maximal class still is a wide open problem. Coclass Conjecture W proposes a way to approach such a classification: It suggests that the coclass graph G associated with the p-groups of maximal class can be determined from a finite subgraph using certain periodic patterns. Here we consider the subgraph G of G associated with those p-groups of maximal class whose automorphism group orders are divisible by p-1. We describe the broad structure of G by determining its so-called skeleton.We investigate the smallest interesting case p = 7 in more detail using computational tools, and propose an explicit version of Conjecture W for G for arbitrary p ≥ 7. Our results are the first explicit evidence in support of Conjecture W for a coclass graph of infinite width.

Original languageEnglish
Pages (from-to)227-256
Number of pages30
JournalJournal of Group Theory
Volume20
Issue number2
DOIs
Publication statusPublished - 2017

Cite this

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Finite p-groups of maximal class with 'large' automorphism groups. / Dietrich, Heiko; Eick, Bettina.

In: Journal of Group Theory, Vol. 20, No. 2, 2017, p. 227-256.

Research output: Contribution to journalArticleResearchpeer-review

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AU - Eick, Bettina

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AB - The classification of p-groups of maximal class still is a wide open problem. Coclass Conjecture W proposes a way to approach such a classification: It suggests that the coclass graph G associated with the p-groups of maximal class can be determined from a finite subgraph using certain periodic patterns. Here we consider the subgraph G of G associated with those p-groups of maximal class whose automorphism group orders are divisible by p-1. We describe the broad structure of G by determining its so-called skeleton.We investigate the smallest interesting case p = 7 in more detail using computational tools, and propose an explicit version of Conjecture W for G for arbitrary p ≥ 7. Our results are the first explicit evidence in support of Conjecture W for a coclass graph of infinite width.

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