Finite groups with all involutions fused

Abstract

We characterize finite groups whose involutions form a single fusion class. In the solvable case, we show that the $2$-length is $1$, extending Thompson's theorem which states that a solvable group with all involutions conjugate has $2$-length $1$. For a nonsolvable group $G$, we prove that if its involutions are fused then they are conjugate; moreover, $G/\mathbf{O}_{2'}(G)$ is almost quasisimple. As an application, we obtain a restriction on $m$ for which an $m$-pyramidal Kirkman triple system can be realized under a nonsolvable group.