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 2 H, if xg 2 H for any
g 2 G, then g 2 H.’ It is proved that G is isomorphic to LK, an extension of an Abelian
2-group L by a group K isomorphic to either A5 for n = 5; or A6 or A7 for n = 6. If G
is simple, we establish that G is isomorphic to A5 for n = 5; or G is isomorphic to A6 or A7 for n = 6.