On partial $\mathrm{\Pi }$-property of subgroups of finite groups

Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies partial $\mathrm{\Pi }$-property in $G$ if there exists a chief series ${\Gamma }_G:1=G_0<G_1<\cdots <G_n=G$ of $G$ such that for every $G$-chief factor $G_i\mathrm{/}{G}_{i-1}$ ($1\le i\le n$) of ${\Gamma }_G$, $\mathrm{\mid }G\mathrm{/}{G}_{i-1}:N_{G\mathrm{/}{G}_{i-1}}(H{G}_{i-1}\mathrm{/}{G}_{i-1}\cap G_i\mathrm{/}{G}_{i-1})\mathrm{\mid }$ is a $\pi (H{G}_{i-1}\mathrm{/}{G}_{i-1}\cap G_i\mathrm{/}{G}_{i-1})$-number. Our main results are listed here: Theorem A. Let $\mathfrak{F}$ be a solubly saturated formation containing $\mathfrak{U}$ and $E$ a normal subgroup of $G$ with $G\mathrm{/}E\in \mathfrak{F}$. Let $X⊴G$ such that $F_p^{\ast }(E)\le X\le E$. Suppose that for any Sylow $p$-subgroup $P$ of $X$, every maximal subgroup of $P$ satisfies partial $\mathrm{\Pi }$-property in $G$. Then one of the following holds: (1) $G\in {\mathfrak{G}}_{p^{\mathrm{\prime }}}\mathfrak{F}$. (2) $X\mathrm{/}{O}_{p^{\mathrm{\prime }}}(X)$ is a quasisimple group with Sylow $p$-subgroups of order $p$. In particular, if $X=F_p^{\ast }(E)$, then $X\mathrm{/}{O}_{p^{\mathrm{\prime }}}(X)$ is a simple group. Theorem B. Let $\mathfrak{F}$ be a solubly saturated formation containing $\mathfrak{U}$ and $E$ a normal subgroup of $G$ with $G\mathrm{/}E\in \mathfrak{F}$. Suppose that for any Sylow $p$-subgroup $P$ of $F_p^{\ast }(E)$, every cyclic subgroup of $P$ of prime order or order 4 (when $P$ is not quaternion-free) satisfies partial $\mathrm{\Pi }$-property in $G$. Then $G\in \mathfrak{G}_{p’}\mathfrak{F}$.