On $\Phi$-$\tau$-Supplement Subgroups of Finite Groups
Received:June 13, 2016  Revised:December 07, 2016
Key Word: Sylow subgroups   subnormal subgroups   subgroup functor   $p$-nilpotent group   $\Phi$-$\tau$-supplement
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11371335).
 Author Name Affiliation Xiaojian MA School of Mathematics and Computer, University of Datong of Shanxi, Shanxi 037009, P. R. China Yuemei MAO School of Mathematics and Computer, University of Datong of Shanxi, Shanxi 037009, P. R. China; School of Mathematical Sciences, University of Science and Technology of China, Anhui 230026, P. R. China
Hits: 269
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 $G$.