Finite groups whose prime graphs do not contain triangles. II

Abstract

The study of finite groups whose prime graphs do not contain triangles is continued. The main result of this paper is the following theorem: if G is a finite nonsolvable group whose prime graph contains no triangles and S(G) is the greatest solvable normal subgroup of G, then |π(G)| ≤ 8 and |π(S(G))| ≤ 3. A detailed description of the structure of a group G satisfying the conditions of the theorem is obtained in the case when π(S(G)) contains a number that does not divide the order of the group G/S(G). We also construct an example of a finite solvable group of Fitting length 5 whose prime graph is a 4-cycle. This completes the determination of the exact bound for the Fitting length of finite solvable groups whose prime graphs do not contain triangles.