On the $IC$-$Π$-property of subgroups of finite groups

Abstract

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the $ Π$-property in $ G $ if $ | G/K : N_{G/K}((H \cap L)K/K)| $ is a $ π(( H \cap L)K/K ) $-number for any chief factor $ L/K $ of $ G $; and we call that $ H $ satisfies the $ IC $-$ Π$-property in $ G $ if $ H\cap [H, G] $ satisfies the $ Π$-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 $IC$-$ Π$-property of some $ p $-subgroups.