Special blocks of finite groups

Abstract

We first determine in this paper the structure of the generalized Fitting subgroup F* (G) of the finite groups G all of whose defect groups (of blocks) are conjugate under the automorphism group Aut(G) to either a Sylow p-subgroup or a fixed p-subgroup of G. Then we prove that if a finite group L acts transitively on the set of its proper Sylow p-intersections, then either L/Op(L) has a T.I. Sylow p-subgroup or p = 2 and the normal closure of a Sylow 2-subgroup of L/O2(L) is 2-nilpotent with completely descripted structure. This solves a long-open problem. We also obtain some generalizations of the classic results by Isaacs and Passman on half-transitivity.