Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the partial $ \mathscr L $-$ \Pi $-property in $ G $ if $ H\unlhd G $, or if $ | G / K : \mathrm{ N} _{ G / K} (HK/K)| $ is a $ \pi (HK/K) $-number for any $ G $-chief factor of type $ H^{ G}/K $ with $ H_{ G}\leq K $. In this paper, we investigate the structure of finite groups under the assumption that some subgroups of prime power order satisfy the partial $ \mathscr L $-$ \Pi $-property.
Categories: math.GR