A lower-bound for the number of conjugacy classes of $A_n$

Abstract

We establish a sharp lower-bound for the number of conjugacy classes $k(A_n)$ in the alternating group $A_n$, for $n\ge 3$. Namely, we show that $k\left(A_n\right)\ge \frac{k\left(A_7\right)}{{\log }_2|A_7|}\cdot {\log }_2\left|A_n\right|$ with equality if, and only if, $n=7$. The observations leading towards this result were obtained through a genetic algorithm developed to search for groups having certain properties.

arXiv comment5 pages