19.36 (2018)

Let $G$ be a periodic group and let $\mathscr{I} = \{x \in G \mid x^2 = 1 \neq x\}$ be the set of its involutions. Let $D$ be a non-empty set of odd integers greater than 1; then $G$ is called a group with $D$-involutions if $G = \langle \mathscr{I} \rangle$ and for $x, y \in \mathscr{I}$ the order of $xy$ is in the set $\{1, 2\} \cup D$ and all these values actually occur. It is clear that if $G$ is a group with $D$-involutions, then $\mathscr{I}$ is a single conjugacy class.

Conjecture: If $G$ is a group with $\{3, 5\}$-involutions, then $G \cong A_5$ or $G \cong PSU(3, 4)$.