Finite groups with conjugacy classes of size $2p^n$

Abstract

We extend Burnside's $p^a$-Theorem on conjugacy classes of prime-power size by proving that no non-abelian finite simple group possesses a conjugacy class of size $2p^n$ for any prime $p$ and $n\ge 0$. Based on this, we partially extend a well-known result of L. Kazarin by showing that, given a finite group $G$, the subgroup generated by a conjugacy class $x^G$ of size $2p^n$ where $x$ has odd order, is a solvable normal subgroup of $G$. If instead the subgroup $⟨x^G⟩$ generated by such a class $x^G$ is non-solvable (which forces $x$ to have even order), then $p=5$ and $⟨x^G⟩/S(⟨x^G⟩)$ is isomorphic to the symmetric group $S_5$, where $S(⟨x^G⟩)$ denotes the solvable radical of $⟨x^G⟩$.