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Journals / Hacettepe Journal of Mathematics and Statistics / 2019 / Cilt: 48 - Sayı: 1

A note on the embedding properties of p-subgroups in finite groups

Xiuyun GUO
Boru Zhang
Pages
102–111
Publication Date
2022-10-18
DOI
—
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AI Summary & Key Takeaways

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⚡ TL;DR(Too Long; Didn't Read)

In this note, we prove that a finite group G is p-supersolvable if and only if there exists a power d of p with p2 ≤ d d if there exists a power d of p with 1 ≤ d < |P| such that H ∩Op(G∗ p2) is normal in Op(G) for all non-meta-cyclic normal subgroups H of P with |H|= d.

📌 3 Key Takeaways

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Objective & Hypothesis:In this note, we prove that a finite group G is p-supersolvable if and only if there exists a power d of p with p2 ≤ d d if there exists a power d of p with 1 ≤ d < |P| such that H ∩Op(G∗ p2) is normal in Op(G) for all non-meta-cyclic normal subgroups H of P with |H|= d.
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Key Findings & Data:In this note, we prove that a finite group G is p-supersolvable if and only if there exists a power d of p with p2 ≤ d d if there exists a power d of p with 1 ≤ d < |P| such that H ∩Op(G∗ p2) is normal in Op(G) for all non-meta-cyclic normal subgroups H of P with |H|= d.
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Conclusion & Impact:In this note, we prove that a finite group G is p-supersolvable if and only if there exists a power d of p with p2 ≤ d d if there exists a power d of p with 1 ≤ d < |P| such that H ∩Op(G∗ p2) is normal in Op(G) for all non-meta-cyclic normal subgroups H of P with |H|= d.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters

Abstract

In this note, we prove that a finite group G is p-supersolvable if and only if there exists a power d of p with p2 ≤ d d if there exists a power d of p with 1 ≤ d < |P| such that H ∩Op(G∗ p2) is normal in Op(G) for all non-meta-cyclic normal subgroups H of P with |H|= d.

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Cite This Article

GUO, X., Zhang, B. (2022). A note on the embedding properties of p-subgroups in finite groups. Hacettepe Journal of Mathematics and Statistics, Cilt: 48 - Sayı: 1, 102–111. https://ajindex.com/articles/529754

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References (12)

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