Characterization of A8 by 3-Centralizers

Let G be a finite group containing a subgroup H isomorphic to an alternating group, An, such that G satisfies the 3-cycle property, namely ’for a 3-cycle x∈H, if xg ∈H for any g ∈G, then g ∈H.’It is proved that for n = 8, G is isomorphic to LK, an extension of an elementary Abelian 2-group L by a group K isomorphic to either A8 or SL(5,2). If G is simple, it is shown that G is isomorphic to A8 or SL(5,2)