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). 

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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 $G$. 
Citation: 
DOI:10.3770/j.issn:20952651.2017.03.005 
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