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]\leq \Phi (S)S_{cG} $ for some $ CAP $-subgroup $ S_{cG}\leq S $ of $ G $, we define $ S $ as an $ IC_{\Phi c} $-subgroup of $ G $. This paper investigates how $ IC_{\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\unlhd G $ be a $ p $-group of order $ p^{n} $. Suppose there exists an integer $ k $ with $ 1\leq k < n $ such that: (i) all subgroups of $ P $ of order $ p^k $ are $ IC_{\Phi c} $-subgroups of $ G $; (ii) if $ p^k = 2 $, then all subgroups of order $ 4 $ are also $ IC_{\Phi c} $-subgroups of $ G $. Then $ P\leq Z_{\mathfrak{U}}(G) $. (2) Given a solvably saturated formation $ \mathfrak{F} \supseteq \mathfrak{U} $, let $ N \unlhd G $ with $ G/N \in \mathfrak{F} $. Suppose for each non-cyclic $ P\in Syl_p(F^{*}(N)) $, where $ p $ is an arbitrary prime in $ \pi (F^{*}(N)) $, the conditions of (1) hold. Then $ G \in \mathfrak{F} $.
Citation: Huajie Zheng, Yong Xu, Songtao Guo. On $ IC_{\Phi c} $-subgroups of finite groups[J]. AIMS Mathematics, 2025, 10(10): 23337-23345. doi: 10.3934/math.20251035
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]\leq \Phi (S)S_{cG} $ for some $ CAP $-subgroup $ S_{cG}\leq S $ of $ G $, we define $ S $ as an $ IC_{\Phi c} $-subgroup of $ G $. This paper investigates how $ IC_{\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\unlhd G $ be a $ p $-group of order $ p^{n} $. Suppose there exists an integer $ k $ with $ 1\leq k < n $ such that: (i) all subgroups of $ P $ of order $ p^k $ are $ IC_{\Phi c} $-subgroups of $ G $; (ii) if $ p^k = 2 $, then all subgroups of order $ 4 $ are also $ IC_{\Phi c} $-subgroups of $ G $. Then $ P\leq Z_{\mathfrak{U}}(G) $. (2) Given a solvably saturated formation $ \mathfrak{F} \supseteq \mathfrak{U} $, let $ N \unlhd G $ with $ G/N \in \mathfrak{F} $. Suppose for each non-cyclic $ P\in Syl_p(F^{*}(N)) $, where $ p $ is an arbitrary prime in $ \pi (F^{*}(N)) $, the conditions of (1) hold. Then $ G \in \mathfrak{F} $.
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Huajie Zheng, Yong Xu, Songtao Guo. On $ IC_{\Phi c} $-subgroups of finite groups[J]. AIMS Mathematics, 2025, 10(10): 23337-23345. doi: 10.3934/math.20251035
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