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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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1435-5337
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Volume 29, Issue 2

# Finite groups have more conjugacy classes

Barbara Baumeister
/ Attila Maróti
/ Hung P. Tong-Viet
Published Online: 2016-05-27 | DOI: https://doi.org/10.1515/forum-2015-0090

## Abstract

We prove that for every $ϵ>0$ there exists a $\delta >0$ such that every group of order $n\ge 3$ has at least $\delta {\mathrm{log}}_{2}n/{\left({\mathrm{log}}_{2}{\mathrm{log}}_{2}n\right)}^{3+ϵ}$ conjugacy classes. This sharpens earlier results of Pyber and Keller. Bertram speculates whether it is true that every finite group of order n has more than ${\mathrm{log}}_{3}n$ conjugacy classes. We answer Bertram’s question in the affirmative for groups with a trivial solvable radical.

MSC 2010: 20E45; 20D06; 20P99

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The GAP Group, GAP–Groups, Algorithms, and Programming, Version 4.7.5, 2014, www.gap-system.org.

Revised: 2016-02-16

Published Online: 2016-05-27

Published in Print: 2017-03-01

The second author was supported by an Alexander von Humboldt Fellowship for Experienced Researchers, by MTA Rényi “Lendület” Groups and Graphs Research Group, and by OTKA grants K84233 and K115799. Tong-Viet’s work is based on the research supported in part by the National Research Foundation of South Africa (Grant Number 93408). Part of the work was done while the last author held a position at the CRC 701 within the project C13 ‘The geometry and combinatorics of groups’. The first and second authors also wish to thank the CRC 701 for its support.

Citation Information: Forum Mathematicum, Volume 29, Issue 2, Pages 259–275, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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