9.28 (1984)

Suppose that a finite group $G$ is generated by a conjugacy class $D$ of involutions and let $D_i = \{d_1 \cdots d_i \mid d_1, \dots, d_i$ are different pairwise commuting elements of $D\}$. What is $G$, if $D_1, \dots, D_n$ are all its different conjugacy classes of involutions? For example, the Fischer groups $F_{22}$ and $F_{23}$ satisfy this condition with $n = 3$.