Finite Groups with Weakly $$\mathcal {H}C$$-Embedded Subgroups

Abstract

A subgroup H of a finite group G is said to be weakly $$\mathcal {H}C$$ -embedded in G if there exists a normal subgroup T of G such that $$H^G = HT$$ and $$H^g \cap N_T(H) \le H$$ for all $$g \in G$$ , where $$H^G$$ is the normal closure of H in G. In this paper, we investigate the structure of the finite group G under the assumption that P a Sylow p-subgroup of G, where p is a prime dividing the order of G, and we fix a subgroup of P of order d with $$1< d < \left| P\right| $$ such that every subgroup H of P of order $$p^nd ~(n = 0, 1)$$ is weakly $$\mathcal {H}C$$ -embedded in G.