Abstract: Let G be a finite group, and let be a set of primes. A subgroup H of G is calledπ-S-quasi-normal in G if H is permuted with every Sylow p-subgroup of G for every p inπ. G is called a Bpgroup if the existence of normal p-complements of N (P) implies that G itself as a normal p-complement, where PE Sylp G . In this paper, we have proved that G is a Bpgroup if G satisfies one of the following conditions: (1) each maximal subgroup of a Sylow p-subgroup P of G is P’-S-quasi-mormal in G ; (2) each second maximal subguoup of a Sylow p- subgroup P of G is p’-S-quasi-normal in G .

Key words: Finite groups, Sylow subgroups, Normal subgroups