Maps, simple groups, and arc-transitive graphs
Abstract
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 $\Gamma$ admitting an almost simple arc-transitive group $X$ of automorphisms, such that $\Gamma$ 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2024
- DOI:
- 10.48550/arXiv.2405.14287
- arXiv:
- arXiv:2405.14287
- Bibcode:
- 2024arXiv240514287L
- Keywords:
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- Mathematics - Group Theory;
- Mathematics - Combinatorics;
- 20B25;
- 20D06;
- 20D08;
- 05C25
- E-Print:
- 48 pages (including 6 pages of results tables at the end)