Subgroups generated by small classes in finite groups

Abstract:

Let $M(G)$ be the subgroup of $G$ generated by all elements that lie in conjugacy classes of the two smallest sizes. Avinoam Mann showed that if $G$ is nilpotent, then $M(G)$ has nilpotence class at most $3$. Using a slight variation on Mann’s methods, we obtain results that do not require us to assume that $G$ is nilpotent. We show that if $G$ is supersolvable, then $M(G)$ is nilpotent with class at most $3$, and in general, the Fitting subgroup of $M(G)$ has class at most $4$.