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 $n \geq 3$. Namely, we show that $k\left(A_n\right) \geq \frac{k\left(A_7\right)}{\log_2\left|A_7\right|} \cdot \log_2\left|A_n\right|$ 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.