On partial $\Pi$-property of subgroups of finite groups

Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies partial $\Pi$-property in $G$ if there exists a chief series $\mathit{\Gamma}_G:1=G_0<G_1<\cdots<G_n=G$ of $G$ such that for every $G$-chief factor $G_i/G_{i-1}$ ($1\leq i\leq n$) of $\mathit{\Gamma}_G$, $|G/G_{i-1}:N_{G/G_{i-1}}(HG_{i-1}/G_{i-1}\cap G_i/G_{i-1})|$ is a $\pi(HG_{i-1}/G_{i-1}\cap G_i/G_{i-1})$-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 $G/E\in \mathfrak{F}$. Let $X\unlhd G$ such that $F_p^*(E)\leq X\leq E$. Suppose that for any Sylow $p$-subgroup $P$ of $X$, every maximal subgroup of $P$ satisfies partial $\Pi$-property in $G$. Then one of the following holds: (1) $G\in \mathfrak{G}_{p'}\mathfrak{F}$. (2) $X/O_{p'}(X)$ is a quasisimple group with Sylow $p$-subgroups of order $p$. In particular, if $X=F_p^*(E)$, then $X/O_{p'}(X)$ 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 $G/E\in \mathfrak{F}$. Suppose that for any Sylow $p$-subgroup $P$ of $F_p^*(E)$, every cyclic subgroup of $P$ of prime order or order 4 (when $P$ is not quaternion-free) satisfies partial $\Pi$-property in $G$. Then $G\in \mathfrak{G}_{p'}\mathfrak{F}$.