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FINITE $p$-GROUPS WITH SMALL AUTOMORPHISM GROUP

Part of: Representation theory of groups

Published online by Cambridge University Press:  20 April 2015

JON GONZÁLEZ-SÁNCHEZ  and
ANDREI JAIKIN-ZAPIRAIN
JON GONZÁLEZ-SÁNCHEZ
Affiliation:
Departamento de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco-Euskal Herriko Unibertsitatea, Apartado 644 48080 Bilbao, Spain
ANDREI JAIKIN-ZAPIRAIN
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, and Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, 28049-Madrid, Spain; [email protected]

Abstract

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For each prime $p$ we construct a family $\{G_{i}\}$ of finite $p$-groups such that $|\text{Aut}(G_{i})|/|G_{i}|$ tends to zero as $i$ tends to infinity. This disproves a well-known conjecture that $|G|$ divides $|\text{Aut}(G)|$ for every nonabelian finite $p$-group $G$.

MSC classification

Primary: 20D15: Nilpotent groups, $p$-groups
Secondary: 20D45: Automorphisms
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 3 , 2015 , e7
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2015

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