STUDIES ON DEFECT-GROUPS

The main results of this paper are as follows: THEOREM 1. If G is a finite group with a strongly p-embedded subgroup then G has a p-block of defect zero. The theorem solves a problem of Alperin. THEOREM 5. Let G be a finite group and D, a p-subgroup of G such that NG(D)ID has a strongly p-embedded subgroup, then D is a defect group for some p-block of G if and only if there exists a p'-element x in G such that D is a Sylow p-subgroup of C(G)(x). COROLLARY 6. If G is a finite group with am abelian Sylow p-subgroup then every strong p-subgroup of G is a defect group for some p-block of G. In particular every maximal Sylow p-intersection of G is a defect group. (C) 1994 Academic Press, Inc.MathematicsSCI(E)4ARTICLE2310-31616