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NEW REDUCTIONS AND LOGARITHMIC LOWER BOUNDS FOR THE NUMBER OF CONJUGACY CLASSES IN FINITE GROUPS

Part of: Representation theory of groups

Published online by Cambridge University Press:  07 September 2012

EDWARD A. BERTRAM
EDWARD A. BERTRAM*
Affiliation:
Department of Mathematics, University of Hawaii, Honolulu, HI 96822, USA (email: ed@math.hawaii.edu)
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Abstract

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The unsolved problem of whether there exists a positive constant $c$ such that the number $k(G)$ of conjugacy classes in any finite group $G$ satisfies $k(G) \geq c \log _{2}|G|$ has attracted attention for many years. Deriving bounds on $k(G)$ from (that is, reducing the problem to) lower bounds on $k(N)$ and $k(G/N)$, $N\trianglelefteq G$, plays a critical role. Recently Keller proved the best lower bound known for solvable groups:

\[ k(G)\gt c_{0} \frac {\log _{2}|G|} {\log _{2} \log _{2} |G|}\quad (|G|\geq 4) \]
using such a reduction. We show that there are many reductions using $k(G/N) \geq \beta [G : N]^{\alpha }$ or $k(G/N) \geq \beta (\log [G : N])^{t}$ which, together with other information about $G$ and $N$ or $k(N)$, yield a logarithmic lower bound on $k(G)$.

Keywords

number of conjugacy classessolvable groups

MSC classification

Secondary: 20D60: Arithmetic and combinatorial problems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 3 , June 2013 , pp. 406 - 424
Copyright
Copyright © 2012 Australian Mathematical Publishing Association Inc. 

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