FINITE GROUPS WITH SOME CAP-SUBGROUPS

Let G be a finite group. A subgroup H of G is called a CAP-subgroup if the following condition is satisfied: for each chief factor K/L of G either. HK = HL or H boolean AND K = H boolean AND L. Let p be a prime factor of vertical bar G vertical bar and let P be a Sylow p-subgroup of G. If d is the minimum number of generators of P then there exists a family of maximal subgroups of?, denoted by M-d(P)={P-1, P-2, ... P-d} such that boolean AND(d)(i=1) P-i = Phi(P). In this paper, we investigate the group G satisfying the condition: every member of a fixed Md(P) is a CAP-subgroup of G. For example, it in addition, G is p-solvable, then G is p-supersolvable.