Finite groups whose minimal subgroups are weakly \mathcal{H}^{\ast}-subgroups

Let G be a finite group‎. ‎A subgroup‎ ‎H of G is called an \mathcal{H}-subgroup in‎ ‎G if N_G(H)\cap H^{g}\leq H for all g\in‎ ‎G. A subgroup H of G is called a weakly‎ ‎\mathcal{H}^{\ast}-subgroup in G if there exists a‎ ‎subgroup K of G such that G=HK and H\cap‎ ‎K is an \mathcal{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 ۴ (if p=۲) is a weakly ‎\mathcal{H}^{\ast}-subgroup in G. Our results improve‎ ‎and extend a series of recent results in the literature‎.

Keywords:

‎weakly mathcal{H}-subgroup , weakly‎ ‎mathcal{H}^{ast}-subgroup‎ , ‎c-supplemented subgroup‎ , ‎generalized‎ ‎Fitting subgroup‎ , ‎saturated formation