On the $IC$-$\Pi$-property of subgroups of finite groups

Abstract

Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies the $\Pi$-property in $G$ if $|G/K:N_{G/K}((H\cap L)K/K)|$ is a $\pi ((H\cap L)K/K)$-number for any chief factor $L/K$ of $G$; and we call that $H$ satisfies the $IC$-$\Pi$-property in $G$ if $H\cap [H,G]$ satisfies the $\Pi$-property in $G$. In this paper, we obtain a criterion of a normal subgroup being contained in the $p\mathfrak{U}$-hypercenter of a finite group by the $IC$-$\Pi$-property of some $p$-subgroups.