On the Prime Graph of a Finite Group

Abstract

Let G be a finite group. We define the prime graph Г(G) of G as follows: The vertices of Г(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. We denote by π(G), the set of all prime divisors of |G|. The degree deg(p) of a vertex p of Г(G) is the number of edges incident with p. If π(G) = {p1, p2,..., pk} where p1< p2 < pk, then we define D(G) = (deg(p1),deg(p2),...,deg(pk)), which is called the degree pattern of G. Given a finite group M, if the number of non-isomorphic groups G such that |G| = |M| and D(G) = D(M) is equal to r, then M is called r-fold OD-characterizable. Also a 1-fold OD-characterizable group is simply called OD-characterizable. In this paper we give some results on characterization of finite groups by prime graphs and OD-characterizability of finite groups. In particular we apply our results to show that the simple groups G2(7), B3(5), A11, and A19are OD-characterizable.