𝑝-parts of stabilizers in primitive permutation groups

Q: What does this research mean for the field?

For a finite solvable primitive permutation group G on a set Ω where p^2 divides |G| for a prime p, there exists a subset of Ω whose stabilizer S has a p-part strictly between 1 and the p-part of G. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

David Gluck

doi:10.1515/jgth-2026-0060

July 16, 2026Journal of Group Theory

𝑝-parts of stabilizers in primitive permutation groups

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Key Points

This work investigates the properties of stabilizers in finite primitive permutation groups, particularly focusing on solvable groups.
Analyzed the structure of finite primitive permutation groups and their stabilizers.
Used a counting argument to explore properties of solvable and non-solvable groups.
Confirmed that for solvable groups, a subset's stabilizer exists, satisfying 1<|S|_p<|G|_p.
Established preliminary counting techniques to analyze stabilizers in non-solvable groups.

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Implication

Analyzes stabilizers' characteristics in solvable groups, suggesting new counting methods.

Abstract

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (G,Ω) be a finite primitive permutation group. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>p</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> p² divide <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> </m:math> G , for a prime 𝑝. We show that, when 𝐺 is solvable, there exists a subset of Ω whose stabilizer 𝑆 has the property that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo><</m:mo> <m:msub> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>S</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>p</m:mi> </m:msub> <m:mo><</m:mo> <m:msub> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>p</m:mi> </m:msub> </m:mrow> </m:math> 1< Sₚ< Gₚ . We offer a counting argument which should be helpful when 𝐺 is not solvable.

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David Gluck (2026) studied this question.

synapsesocial.com/papers/6a5b187518557b26c203a479 https://doi.org/10.1515/jgth-2026-0060

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𝑝-parts of stabilizers in primitive permutation groups

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Implication

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𝑝-parts of stabilizers in primitive permutation groups

Key Points

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Implication

Abstract

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Cite This Study

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