On $\mathfrak{F}_τ$-$s$-Supplemented Subgroups of Finite Groups (Journal of Mathematical Study, 2018)
Let $\mathfrak{F}$ be a non-empty formation of groups, $τ$ a subgroup functor and $H$ a $p$-subgroup of a finite group $G.$ Let $\overline{G}=G/H_G$ and $\overline{H} =H/H_G.$ We say that $H$ is $\mathfrak{F}_τ$-$s$-supplemented in $G$ if for some subgroup $\overline{T}$ and some $τ$-subgroup $\overline{S}$ of $\overline{G}$ containedin $\overline{H},$ $\overline{H}\overline{T}$ is subnormal in $\overline{G}$ and $\overline{H} ∩ \overline{T} ≤ \overline{S}Z_{\mathfrak{F}}(\overline{G}).$ In this paper, we investigate theinfluence of $\mathfrak{F}_τ$-$s$-supplemented subgroups on the structure of finite groups. Somenew characterizations about solubility of finite groups are obtained.
Keywords: Subnormal subgroup; subgroup functor; soluble group.