On the structure of locally finite groups with small centralizers

Let A stand for the group isomorphic with S4, the symmetric group on four symbols. Let V denote the normal subgroup of order four in A and choose an involution α∈A∖V. The Sylow 2-subgroup D of A is V〈α〉 and this is isomorphic with the dihedral group of order 8. We prove that if G is a locally finite group containing a subgroup isomorphic with D such that CG(V) is finite and CG(α) has finite exponent, then [G,D]′ has finite exponent. If G is a locally finite group containing a subgroup isomorphic with A such that CG(V) is finite and CG(α) has finite exponent, then G has finite exponent.