Characterization of An for n = 5, 6 by 3-centralizers

Abstract

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.