A note on groups generated by involutions and sharply $2$-transitive groups

Abstract:

Let $G$ be a group generated by a set $C$ of involutions which is closed under conjugation. Let $\pi$ be a set of odd primes. Assume that either (1) $G$ is solvable, or (2) $G$ is a linear group.

We show that if the product of any two involutions in $C$ is a $\pi$-element, then $G$ is solvable in both cases and $G=O_{\pi }(G)\langle t\rangle$, where $t\in C$.

If (2) holds and the product of any two involutions in $C$ is a unipotent element, then $G$ is solvable.

Finally we deduce that if $\mathcal {G}$ is a sharply $2$-transitive (infinite) group of odd (permutational) characteristic, such that every $3$ involutions in $\mathcal {G}$ generate a solvable or a linear group; or if $\mathcal {G}$ is linear of (permutational) characteristic $0,$ then $\mathcal {G}$ contains a regular normal abelian subgroup.