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 2H, if xg 2H for
any g 2 G, then g 2 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).