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# Journal of Group Theory

Editor-in-Chief: Parker, Christopher W.

Managing Editor: Wilson, John S. / Khukhro, Evgenii I. / Kramer, Linus

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1435-4446
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Volume 20, Issue 2

# Finite p-groups of maximal class with ‘large’ automorphism groups

Heiko Dietrich
/ Bettina Eick
Published Online: 2016-09-14 | DOI: https://doi.org/10.1515/jgth-2016-0036

## 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 $\mathcal{𝒢}$ associated with the p-groups of maximal class can be determined from a finite subgraph using certain periodic patterns. Here we consider the subgraph ${\mathcal{𝒢}}^{\ast }$ of $\mathcal{𝒢}$ associated with those p-groups of maximal class whose automorphism group orders are divisible by $p-1$. We describe the broad structure of ${\mathcal{𝒢}}^{\ast }$ 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 ${\mathcal{𝒢}}^{\ast }$ for arbitrary $p\ge 7$. Our results are the first explicit evidence in support of Conjecture W for a coclass graph of infinite width.

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Published Online: 2016-09-14

Published in Print: 2017-03-01

This research was supported by a Go8-DAAD Joint Research Co-operation Scheme, project “Groups of Prime-Power Order and Coclass Theory”. The second author was also supported by an ARC DECRA (Australia), project DE140100088.

Citation Information: Journal of Group Theory, Volume 20, Issue 2, Pages 227–256, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883,

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