Bounds on the number of conjugacy classes of the symmetric and alternating groups

Bret Benesh, Cong Tuan Son Van

Let $G$ be a finite group with Sylow subgroups $P_1,\dots ,P_n$, and let $k(G)$ denote the number of conjugacy classes of $G$. Pyber asked if $k(G)\le {\prod }_{i=1}^{n}k(P_i)$ for a…