The solution to Kourovka problem 21.88

Abstract

We give a negative answer to Kourovka Notebook Problem 21.88: no finite group of odd order has commuting probability $1/17$. This follows from a structural theorem asserting that, whenever $p$ is an odd prime and $cp(G)=1/p$, a Sylow $p$-subgroup of $G$ is normal and abelian. Together with Burnside's congruence for the number of conjugacy classes of a group of odd order, this also excludes $cp(G)=1/p$ for every odd prime $p<97$. We further study the next unresolved case not excluded by this congruence, namely $p=97$.