On $\Phi$-$\tau$-Supplement Subgroups of Finite Groups

Abstract:

      Let $\tau$ be a subgroup functor and $H$ a $p$-subgroup of a finite group $G$. Let $\bar{G}=G/H_{G}$ and $\bar{H}=H/H_{G}$. We say that $H$ is $\Phi$-$\tau$-supplement in $G$ if $\bar{G}$ has a subnormal subgroup $\bar{T}$ and a $\tau$-subgroup $\bar{S}$ contained in $\bar{H}$ such that $\bar{G}=\bar{H}\bar{T}$ and $\bar{H}\cap\bar{T}\leq \bar{S}\Phi(\bar{H})$. In this paper, some new characterizations of hypercyclically embedability and $p$-nilpotency of a finite group are obtained based on the assumption that some primary subgroups are $\Phi$-$\tau$-supplement in $G$.