On locally finite groups with a four-subgroup whose centralizer is small

Let $G$ be a locally finite group which contains a non-cyclic subgroup $V$ of order four such that $C_{G}left( Vright) $ is finite and $C_{G}left( phi right)$ has finite exponent for some $phi in V$ . We show that $[G,phi ]^{prime }$ has finite exponent. This enables us to deduce that $G$ has a normal series $1le G_1le G_2le G_3le G$ such that $G_1$ and $G/G_2$ have finite exponents while $G_2/G_1$ is abelian. Moreover $G_3$ is hyperabelian and has finite index in $G$ .