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Dergiler / 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
Sayfa
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Yayın Tarihi
2026-08-17
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

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 Maddede Bu Makale

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Amaç & Hipotez: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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Temel Bulgular & Veri: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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Sonuç & Katkı: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.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler: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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Bu Makaleye Atıf Yap

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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Kaynakça (8)

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  2. X. Chen and W. Guo, On weakly S-embedded and weakly $\tau$-embedded subgroups, Siberian Math. J. 54, 931–945, 2013.
  3. B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, Heidelberg, New York, 1967.
  4. I. M. Isaacs, Finite Group Theory, American Mathematical Society, Providence, RI, 2008.
  5. H. Kurzweil and B. Stellmacher, The Theory of Finite Groups, Springer-Verlag, Berlin, Heidelberg, 2004.
  6. X. Li and T. Zhao, $S\Phi$-supplemented subgroups of finite groups, Ukrainian Math. J. 64, 102–109, 2012.
  7. H. Meng and A. Ballester-Bolinches, On a paper of Beltrán and Shao about coprime action, J. Pure Appl. Algebra 224, 106303, 2020.
  8. S. Yang and X. Li, Generalized Frattini subgroup and supplements of subgroups of groups, Comm. Algebra 48, 2517–2527, 2020.