Vertex stabilizers of locally s-arc transitive graphs of pushing up type

John van Bon; Chris Parker

doi:10.48550/arXiv.2312.02627

Vertex stabilizers of locally s-arc transitive graphs of pushing up type

John van Bon^*, Chris Parker

^*Corresponding author for this work

Mathematics

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Abstract

Suppose that Δ a thick, locally finite and locally s-arc transitive G-graph with s≥4. For a vertex z in Δ, let G_z be the stabilizer of z and G^[1]_z be the kernel of the action of G_z on the neighbours of z. We say Δ is of pushing up type provided there exists a prime p and a 1-arc (x,y) such that C_Gz(O_p(G^[1]_z))≤O_p(G^[1]_z) for z∈{x,y} and Op(G^[1]_x)≤O_p(G^[1]_y). We show that if Δ is of pushing up type, then O_p(G^[1]_x) is elementary abelian and G_x/G^[1]_x≅X with PSL₂(p^a)≤X≤PΓL₂(p^a).

Original language	English
Journal	Journal of Algebraic Combinatorics
Early online date	8 May 2024
DOIs	https://doi.org/10.48550/arXiv.2312.02627 https://doi.org/10.1007/s10801-024-01326-x
Publication status	E-pub ahead of print - 8 May 2024

Bibliographical note

Publisher Copyright:
© The Author(s) 2024.

Keywords

Locally s-arc transitive graphs
Group amalgams

ASJC Scopus subject areas

Algebra and Number Theory
Discrete Mathematics and Combinatorics

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Cite this

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title = "Vertex stabilizers of locally s-arc transitive graphs of pushing up type",

abstract = "Suppose that Δ a thick, locally finite and locally s-arc transitive G-graph with s≥4. For a vertex z in Δ, let Gz be the stabilizer of z and G[1]z be the kernel of the action of Gz on the neighbours of z. We say Δ is of pushing up type provided there exists a prime p and a 1-arc (x,y) such that CGz(Op(G[1]z))≤Op(G[1]z) for z∈{x,y} and Op(G[1]x)≤Op(G[1]y). We show that if Δ is of pushing up type, then Op(G[1]x) is elementary abelian and Gx/G[1]x≅X with PSL2(pa)≤X≤PΓL2(pa).",

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doi = "10.48550/arXiv.2312.02627",

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AU - Parker, Chris

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N2 - Suppose that Δ a thick, locally finite and locally s-arc transitive G-graph with s≥4. For a vertex z in Δ, let Gz be the stabilizer of z and G[1]z be the kernel of the action of Gz on the neighbours of z. We say Δ is of pushing up type provided there exists a prime p and a 1-arc (x,y) such that CGz(Op(G[1]z))≤Op(G[1]z) for z∈{x,y} and Op(G[1]x)≤Op(G[1]y). We show that if Δ is of pushing up type, then Op(G[1]x) is elementary abelian and Gx/G[1]x≅X with PSL2(pa)≤X≤PΓL2(pa).

AB - Suppose that Δ a thick, locally finite and locally s-arc transitive G-graph with s≥4. For a vertex z in Δ, let Gz be the stabilizer of z and G[1]z be the kernel of the action of Gz on the neighbours of z. We say Δ is of pushing up type provided there exists a prime p and a 1-arc (x,y) such that CGz(Op(G[1]z))≤Op(G[1]z) for z∈{x,y} and Op(G[1]x)≤Op(G[1]y). We show that if Δ is of pushing up type, then Op(G[1]x) is elementary abelian and Gx/G[1]x≅X with PSL2(pa)≤X≤PΓL2(pa).

KW - Locally s-arc transitive graphs

KW - Group amalgams

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DO - 10.48550/arXiv.2312.02627

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Vertex stabilizers of locally s-arc transitive graphs of pushing up type

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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