Finite groups have more conjugacy classes

Abstract

AbstractWe prove that for every ${\epsilon>0}$ there exists a ${\delta>0}$ such that every group of order ${n\geq 3}$ has at least ${\delta\log_{2}n/{(\log_{2}\log_{2}n)}^{3+\epsilon}}$ 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 ${\log_{3}n}$ conjugacy classes. We answer Bertram’s question in the affirmative for groups with a trivial solvable radical.