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

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Abstract
Let G G be a group generated by a set C C of involutions which is closed under conjugation. Let π \pi be a set of odd primes. Assume that either (1) G G is solvable, or (2) G G is a linear group. We show that if the product of any two involutions in C C is a π \pi -element, then G G is solvable in both cases and G = O π ( G ) ⟨ t ⟩ G=O_{\pi }(G)\langle t\rangle , where t ∈ C t\in C . If (2) holds and the product of any two involutions in C C is a unipotent element, then G G is solvable. Finally we deduce that if G \mathcal {G} is a sharply 2 2 -transitive (infinite) group of odd (permutational) characteristic, such that every 3 3 involutions in G \mathcal {G} generate a solvable or a linear group; or if G \mathcal {G} is linear of (permutational) characteristic 0 , 0, then G \mathcal {G} contains a regular normal abelian subgroup.

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