On $I{C}_{\mathrm{\Phi }c}$-subgroups of finite groups

In group theory, the study of how the properties of specific subgroups affect group structure is a research field that remains highly active. For a finite group $G$ and its subgroup $S$, if $S\cap [S,G]\le \mathrm{\Phi }(S)S_{cG}$ for some $CAP$-subgroup $S_{cG}\le S$ of $G$, we define $S$ as an $I{C}_{\mathrm{\Phi }c}$-subgroup of $G$. This paper investigates how $I{C}_{\mathrm{\Phi }c}$-subgroups impact finite group structure, yielding novel theorems that generalize prior work and enrich the theory of group structure. Specifically, we establish the following main theorems: (1) Let $P⊴G$ be a $p$-group of order $p^n$. Suppose there exists an integer $k$ with $1\le k<n$ such that: (i) all subgroups of $P$ of order $p^k$ are $I{C}_{\mathrm{\Phi }c}$-subgroups of $G$; (ii) if $p^k=2$, then all subgroups of order $4$ are also $I{C}_{\mathrm{\Phi }c}$-subgroups of $G$. Then $P\le Z_{\mathfrak{U}}(G)$. (2) Given a solvably saturated formation $\mathfrak{F}\supseteq \mathfrak{U}$, let $N⊴G$ with $G/N\in \mathfrak{F}$. Suppose for each non-cyclic $P\in Sy{l}_p(F^{\ast }(N))$, where $p$ is an arbitrary prime in $\pi (F^{\ast }(N))$, the conditions of (1) hold. Then $G\in \mathfrak{F}$.