ON THE PRIME GRAPH OF A FINITE GROUP.

Let G be a finite group. We define the prime graph Γ(G) of G as follows: The vertices of G(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 p(G), the set of all prime divisors of jGj. The degree deg(p) of a vertex p of G(G) is the number of edges incident with p. If μ(G) = fp1; p2; ... pkg 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 jGj = jMj 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 A19 are OD-characterizable. [ABSTRACT FROM AUTHOR]