# Edward Bertram

Affiliation: University of Hawaii at Manoa

Email: ed@math.hawaii.edu

Title Of Talk: Recent progress on an old unsolved problem in finite group theory

Abstract: In 1903 E.Landau showed why, given a positive integer $k$, only a finite number of finite groups have exactly $k$ conjugacy classes. In the 1960's several authors (including P. Erdös and P.Turan) proved, using only the class-equation, that always $k(G) \gt \log_2 \, \log_2({\lvert{G}\rvert})$. But group-theorists have also proved that for infinite collections of groups, including all supersolvable groups, that $k(G)\gt \log_3 {\lvert{G}\rvert}$, and from the contributions of many authors toward classifying finite groups by their number $k$ of classes (now completed for $k < 15$), we know that whenever ${\lvert{G}\rvert}$ is no more than $3^{15}$, $k(G) \gt \log_3 ({\lvert{G}\rvert})$.

When $G$ is a solvable group, we discuss number-theoretic evidence related to the prime power factorization of ${\lvert{G}\rvert}$, which along with recent developments would yield that $k(G) \gt \log_3 ({\lvert{G}\rvert})$ when $G$ is solvable and ${\lvert{G}\rvert}$ is sufficiently large.