Abstract As is well known, the embedding of all maximal subgroups of P of a group G can determine the structure of G, where p ∈ π (G) and P ∈ Syl p (G). Based on the topic, we defined two sets δ (P, G) = P 1 ⋖ P ∣ P ∩ P, G ≰ P 1 and F (P, G) = P 1 ⋖ P ∣ P G ≰ P 1 F (, ₆) =\{P₁ F (P, G) {F (, ₆) satisfies the ICΦ-property). To some extent, the two sets will have wide application in investigating the p -nilpotency of finite groups and improving some known theorems.
Xiang et al. (Thu,) studied this question.