THE INFLUENCE OF WEAKLY H-SUBGROUPS ON THE STRUCTURE OF FINITE GROUPS

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 epsilon G. A subgroup H of G is called a weakly H-subgroup in G if there exists a normal subgroup K of G such that G = HK and H boolean AND K is an H-subgroup in G. In this article, we investigate the structure of a group G in which every subgroup with order p(m) of a Sylow p-subgroup P of G is a weakly H-subgroup in G, where m is a fixed positive integer. Our results improve and extend the main results of Skiba [13], Jaraden and Skiba [11], Guo and Wei [8], Tong-Veit [15] and Li et al. [12].