Abstract
Let G be a finite group, and H a subgroup of G. We say that H is an H-subgroup in G if N-G (H) boolean AND H-9 <= H for any g is an element of G. We say that H is weakly H-embedded in G if G has a normal subgroup K such that H-G = HK and H boolean AND K is an 14-subgroup in G. For each prime p dividing the order of G, let P be a non-cyclic Sylow p-subgroup of G. We fix a p-power integer d with 1 < d < 111, and study the structure of G under the assumption that each subgroup of P of order d and pd is weakly H-embedded in G. Some new results about the p-nilpotency and supersolvability of G are obtained.