@article{Zheng2025, 
author = {Huajie Zheng and Yong Xu and Songtao Guo},
title = {On    I      C          Φ      c      -subgroups of finite groups},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {10},
pages = {23337-23345},
keywords = {finite group, ICΦc-subgroup, CAP-subgroup, generalized Fitting subgroup, saturated formation},
url = {https://www.sciopen.com/article/10.3934/math.20251035},
doi = {10.3934/math.20251035},
abstract = {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  ∩  [  S  ,  G  ]  ≤  Φ  (  S  )      S          c      G       for some    C  A  P-subgroup        S          c      G        ≤  S of    G, we define    S as an    I      C          Φ      c      -subgroup of    G. This paper investigates how    I      C          Φ      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  ≤  k  <  n such that: (i) all subgroups of    P of order        p    k   are    I      C          Φ      c      -subgroups of    G; (ii) if        p    k    =  2, then all subgroups of order    4 are also    I      C          Φ      c      -subgroups of    G. Then    P  ≤      Z                  U              (  G  ). (2) Given a solvably saturated formation        F    ⊇      U  , let    N  ⊴  G with    G      /    N  ∈      F  . Suppose for each non-cyclic    P  ∈  S  y      l    p    (      F          ∗        (  N  )  ), where    p is an arbitrary prime in    π  (      F          ∗        (  N  )  ), the conditions of (1) hold. Then    G  ∈      F  .}
}