Abstract
LetGGbe a group generated by a setCCof involutions which is closed under conjugation. Letπ\pibe a set of odd primes. Assume that either (1)GGis solvable, or (2)GGis a linear group.We show that if the product of any two involutions inCCis aπ\pi-element, thenGGis solvable in both cases andG=Oπ(G)⟨t⟩G=O_{\pi }(G)\langle t\rangle, wheret∈Ct\in C.If (2) holds and the product of any two involutions inCCis a unipotent element, thenGGis solvable.Finally we deduce that ifG\mathcal {G}is a sharply22-transitive (infinite) group of odd (permutational) characteristic, such that every33involutions inG\mathcal {G}generate a solvable or a linear group; or ifG\mathcal {G}is linear of (permutational) characteristic0,0,thenG\mathcal {G}contains a regular normal abelian subgroup.
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