OD-Characterization of alternating and symmetric groups of degrees 16 and 22

Let G be a finite group and pi(G) be the set of all prime divisors of its order. The prime graph GK(G) of G is a simple graph with vertex set pi(G), and two distinct primes p, q is an element of pi(G) are adjacent by an edge if and only if G has an element of order pq. For a vertex p is an element of pi(G), the degree of p is denoted by deg(p) and as usual is the number of distinct vertices joined to p. If pi(G) = {p(1), p(2),..., p(k)}, where p(1) < p(2) < ... < p(k), then the degree pattern of G is defined by D(G) = (deg(p(1)), deg(p(2)), ..., deg(p(k))). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions vertical bar H vertical bar = vertical bar G vertical bar and D(H) = D(G). In addition, a 1-fold OD-characterizable group is simply called OD-characterizable. In the present article, we show that the alternating group A(22) is OD-characterizable. We also show that the automorphism groups of the alternating groups A(16) and A(22), i.e., the symmetric groups S-16 and S-22 are 3-fold OD-characterizable. It is worth mentioning that the prime graph associated to all these groups are connected.