AN ONC-CHARACTERIZATION OF A\(_{14}\) AND A\(_{15}\)

Abstract

Let G be a finite group, o₁(G) denote the largest element order of G, n₁(G) the number of the elements of order o₁(G).  Assume that G totally has r elements of order o₁(G), whose centralizers have distinct orders, say, they are c₁(G), i = 1, 2, … , r.  The following quantity is called the 1st ONC−degree of G

     ONC₁(G) ={o₁(G); n₁(G); c₁(G), c₂(G), … , c_r(G)},

denoted as ONC₁(G).  It has been proved that K₃−simple groups, L₂(q) (q = 8, 11, 13, 17, 19, 23, 29), Mathieu simple groups, Janko Groups and alternating groups A_n (5 ≤ n ≤ 13) can be characterized by their 1st ONC−degrees, but unfortunately L₂(q) (q=16, 25) cannot be characterized by the 1st ONC−degree.  Since the ONC−degree of an alternating group usually contains only 3 numbers, so it is interesting to study if an alternating group can be characterized by the 1st ONC−degree.  We shall prove that A₁₄ can be characterized by the 1st ONC−degree, but we can not prove A₁₅ does by using our approaches.  We shall prove if the prime graph of G is not connected and ONC₁(G) = ONC1(A₁₅), then G \(\cong\) A₁₅.