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NEW REDUCTIONS AND LOGARITHMIC LOWER BOUNDS FOR THE NUMBER OF CONJUGACY CLASSES IN FINITE GROUPS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 87 / Issue 3 / June 2013
- Published online by Cambridge University Press:
- 07 September 2012, pp. 406-424
- Print publication:
- June 2013
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- Article
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- You have access
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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: using such a reduction. We show that there are many reductions using
\[ k(G)\gt c_{0} \frac {\log _{2}|G|} {\log _{2} \log _{2} |G|}\quad (|G|\geq 4) \]
$k(G/N) \geq \beta [G : N]^{\alpha }$ 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 
$k(G)$.
