ON NONSOLVABLE GROUPS WHOSE PRIME DEGREE GRAPHS HAVE FOUR VERTICES AND ONE TRIANGLE

Abstract

Let G be a finite group. The prime degree graph of G, denoted by Delta(G), is an undirected graph whose vertex set is rho(G) and there is an edge between two distinct primes p and q if and only if pq divides some irreducible character degree of G. In general, it seems that the prime graphs contain many edges and thus they should have many triangles, so one of the cases that would be interesting is to consider those finite groups whose prime degree graphs have a small number of triangles. In this paper we consider the case where for a nonsolvable group G, Delta(G) is a connected graph which has only one triangle and four vertices.