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Let G be a finite group. Moghaddamfar et al. defined prime graph Γ(G) of group G as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge, denoted by p ∼ q , if there is an element in G of order pq . AssumeDenote by π(G) the set of prime divisor of |G| . Let GK(G) be the graph with vertex set π(G) such that two primes p and q in π(G) are joined by an edge if G has an element of order p • q . We set s(G) to denote the number of connected components of the prime graph GK(G) . Some authors proved some groups are OD -characterizable with s(G) ≥ 2 .Then for s(G) = 1 , what is the influence of OD on the structure of groups? We knew that the alternating groups Ap+3 , where 7 ̸ = p ∈ π(100!) , A130 and A140 are OD -characterizable. Therefore, we naturally ask the following question: if s(G) = 1 , then is there a group OD -characterizable? In this note, we give a characterization of Ap+3 except A10 with s(Ap+3) = 1 , by OD , which gives a positive answer to Moghaddamfar and Rahbariyan's conjecture.

Let G be a group and ω(G) be the set of element orders of G. Let k ∈ ω(G) and s k be the number of elements of order k in G. Let nse(G) = {s k | k ∈ ω(G)}. L3(2) ∼ = L2 (7) is uniquely determined by nse(G). In this paper, we prove that if G is a group such that nse(G) = nse (L3(4)), then G ∼ = L3(4).

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Let G be a group. Denote by π(G) the set of prime divisors of |G|. Let GK(G) be the graph with vertex set π(G) such that two primes p and q in π(G) are joined by an edge if G has an element of order p · q. We set s(G) to denote the number of connected components of the prime graph GK(G). Denote by N(G) the set of nonidentity orders of conjugacy classes of elements in G. Alavi and Daneshkhah proved that the groups, A n where n = p, p + 1, p + 2 with s(G) ≥ 2, are characterized by N(G). As a development of these topics, we will prove that if G is a finite group with trivial center and N(G) = N(A p+3) with p + 2 composite, then G is isomorphic to A p+3.