A NOTE ON GROUPS GENERATED BY INVOLUTIONS AND SHARPLY 2-TRANSITIVE GROUPS

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) < t >, where t is an element of 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 G is a sharply 2-transitive (infinite) group of odd (permutational) characteristic, such that every 3 involutions in G generate a solvable or a linear group; or if G is linear of (permutational) characteristic 0, then G contains a regular normal abelian subgroup.