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Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 55 Issue: 4

S$\Phi$-supplemented and coprime action in finite groups

Qiang Liu(Nanjing University of Posts and Telecommunications)0009-0000-6261-1171
Yuxi Huang(Nanjing University of Posts and Telecommunications)0009-0004-4763-2344
Pages
—
Publication Date
2026-08-17
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AI Summary & Key Takeaways

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⚡ TL;DR(Too Long; Didn't Read)

In this paper, we study $S\Phi$-supplement in the coprime action setting so as to obtain several $p$-nilpotency criteria in terms of certain subsets of maximal invariant subgroups of its Sylow subgroups.

📌 3 Key Takeaways

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Objective & Hypothesis:An $A$-invariant subgroup $H$ of $G$ is said to be $AS\Phi$-supplemented in $G$ if there exists an $A$-invariant normal subgroup $T$ of $G$ such that $G=HT$ and $ H\cap T\le \Phi(H)$, where $\Phi(H)$ is the Frattini subgroup of $H$.
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Key Findings & Data:An $A$-invariant subgroup $H$ of $G$ is said to be $AS\Phi$-supplemented in $G$ if there exists an $A$-invariant normal subgroup $T$ of $G$ such that $G=HT$ and $ H\cap T\le \Phi(H)$, where $\Phi(H)$ is the Frattini subgroup of $H$.
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Conclusion & Impact:In this paper, we study $S\Phi$-supplement in the coprime action setting so as to obtain several $p$-nilpotency criteria in terms of certain subsets of maximal invariant subgroups of its Sylow subgroups.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Coprime action · SΦ-supplemented · p-nilpotency

Abstract

An $A$-invariant subgroup $H$ of $G$ is said to be $AS\Phi$-supplemented in $G$ if there exists an $A$-invariant normal subgroup $T$ of $G$ such that $G=HT$ and $ H\cap T\le \Phi(H)$, where $\Phi(H)$ is the Frattini subgroup of $H$. In this paper, we study $S\Phi$-supplement in the coprime action setting so as to obtain several $p$-nilpotency criteria in terms of certain subsets of maximal invariant subgroups of its Sylow subgroups.

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Cite This Article

Liu, Q., Huang, Y. (2026). S$\Phi$-supplemented and coprime action in finite groups. Hacettepe Journal of Mathematics and Statistics, Volume: 55 Issue: 4. https://doi.org/10.15672/hujms.1823576

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References (8)

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