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

Abstract

Let G be a finite group and π(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 π(G), and two distinct primes p, q ∈ π(G) are adjacent by an edge if and only if G has an element of order pq. For a vertex p ∈ π(G), the degree of p is denoted by deg(p) and as usual is the number of distinct vertices joined to p. If π(G) = {p₁, p₂, …, p_k}, where p₁<p₂<…<p_k, then the degree pattern of G is defined by D(G) = (deg(p₁), deg(p₂), … , deg(p_k)). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions H = G and D(H) = D(G). In addition, a 1-fold Odcharacterizable group is simply called OD-characterizable. In the present article, we show that the alternating group A₂₂ is OD-characterizable. We also show that the automorphism groups of the alternating groups A₁₆ and A₂₂, i.e., the symmetric groups S₁₆ and S₂₂ are 3-fold OD-characterizable. It is worth mentioning that the prime graph associated to all these groups are connected.

Key words： OD-characterizability of a finite group; degree pattern; prime graph