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

TL;DR

The paper defines the IC-Π-property, a hybrid condition that ensures the intersection with the commutator subgroup behaves with respect to the Π-property, to study how p-subgroups control the $p\mathfrak{U}$-hypercenter of finite groups. It proves a normal-subgroup criterion: if $N$ is normal in $G$, $p| |N|$, and there exists $X$ with $F_p^{*}(N) \le X \le N$ such that a Sylow $p$-subgroup $P$ of $X$ has all nontrivial subgroups of order $d$ (a $p$-power with $1<d<|P|$) satisfying the IC-$Π$-property (with an extra clause for $p=2$ when $P$ is non-abelian), then $N \le Z_{p\mathfrak{U}}(G)$. The proofs use a minimal counterexample strategy and a cascade of reduction lemmas to derive inclusions in $Z_{p\mathfrak{U}}(G)$, yielding a contradiction. Beyond this core theorem, the work shows that IC-Π-property encompasses many classical embedding properties and yields broad corollaries for formations and hypercenter structure, providing a unifying framework for understanding the internal composition 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.

Theorems & Definitions (35)

• Definition 1
• Example 2
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 2.1: li-2011
• Lemma 2.2
• proof
• Lemma 2.3