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

Abstract

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 all finite groups $G$. With the help of GAP, we prove that Pyber's inequality holds for all symmetric and alternating groups.

3 pages