A Note on ${\mathcal {H}}$C-Subgroups of Finite Groups

Abstract

A subgroup H of a finite group G is said to be an ${\mathcal {H}}$-subgroup in G if $N_{G}(H)\cap H^{g}\le H$ for all $g \in G$; we say that H is an ${\mathcal {H}}C$-subgroup in G if there is a normal subgroup T of G such that $G=HT$ and $H^{g}\cap N_{T}(G)\le H$ for all $g \in G$. In this note, we obtain some interesting results by applying ${\mathcal {H}}C$-subgroups. As applications, many known results are generalized in recent literature.