Characterization of  p -nilpotency and IC-properties of finite groups

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi mathvariant="fraktur">F</m:mi> </m:mrow> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>P</m:mi> <m:mo>,</m:mo> <m:mi>G</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>P</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>⋖</m:mo> <m:mi>P</m:mi> <m:mo stretchy="false">∣</m:mo> <m:msub> <m:mrow> <m:mi>P</m:mi> </m:mrow> <m:mrow> <m:mi>G</m:mi> </m:mrow> </m:msub> <m:mo>≰</m:mo> <m:msub> <m:mrow> <m:mi>P</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> F(P,G)=₁< dotP PG P₁\ , which provide a new way to select “some” maximal subgroups of P instead of “all”. Further, we analyzed p -nilpotency of a group under the condition that every element in δ ( P , G ) satisfies the ICΦ-property (or P is an ICΦ s -subgroup of G and every element in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi mathvariant="fraktur">F</m:mi> </m:mrow> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>P</m:mi> <m:mo>,</m:mo> <m:mi>G</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msub> </m:math> F(P,G) 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.