Abstract
Let G be a finite group. A subgroup H of G is called an H-subgroup in G if N-G (H) boolean AND H-g <= H for all g is an element of G. A subgroup H of G is called a weakly H*-subgroup in G if there exists a subgroup K of G such that G = H K and H boolean AND K is an H-subgroup in G. We investigate the structure of the finite group G under the assumption that every cyclic subgroup of G of prime order p or of order 4 ( if p = 2) is a weakly H*-subgroup in G. Our results improve and extend a series of recent results in the literature.