Maps, simple groups, and arc-transitive graphs
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913359740272640 |
|---|---|
| author | Liebeck, Martin W. Praeger, Cheryl E. |
| author_facet | Liebeck, Martin W. Praeger, Cheryl E. |
| contents | We determine all factorisations $X=AB$, where $X$ is a finite almost simple group and $A,B$ are core-free subgroups such that $A\cap B$ is cyclic or dihedral. As a main application, we classify the graphs $Γ$ admitting an almost simple arc-transitive group $X$ of automorphisms, such that $Γ$ has a 2-cell embedding as a map on a closed surface admitting a core-free arc-transitive subgroup $G$ of $X$. We prove that apart from the case where $X$ and $G$ have socles $A_n$ and $A_{n-1}$ respectively, the only such graphs are the complete graphs $K_n$ with $n$ a prime power, the Johnson graphs $J(n,2)$ with $n-1$ a prime power, and 14 further graphs. In the exceptional case, we construct infinitely many graph embeddings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_14287 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maps, simple groups, and arc-transitive graphs Liebeck, Martin W. Praeger, Cheryl E. Group Theory Combinatorics 20B25, 20D06, 20D08, 05C25 We determine all factorisations $X=AB$, where $X$ is a finite almost simple group and $A,B$ are core-free subgroups such that $A\cap B$ is cyclic or dihedral. As a main application, we classify the graphs $Γ$ admitting an almost simple arc-transitive group $X$ of automorphisms, such that $Γ$ has a 2-cell embedding as a map on a closed surface admitting a core-free arc-transitive subgroup $G$ of $X$. We prove that apart from the case where $X$ and $G$ have socles $A_n$ and $A_{n-1}$ respectively, the only such graphs are the complete graphs $K_n$ with $n$ a prime power, the Johnson graphs $J(n,2)$ with $n-1$ a prime power, and 14 further graphs. In the exceptional case, we construct infinitely many graph embeddings. |
| title | Maps, simple groups, and arc-transitive graphs |
| topic | Group Theory Combinatorics 20B25, 20D06, 20D08, 05C25 |
| url | https://arxiv.org/abs/2405.14287 |