{ "id": "2502.01287", "version": "v1", "published": "2025-02-03T11:58:56.000Z", "updated": "2025-02-03T11:58:56.000Z", "title": "Kronecker classes and cliques in derangement graphs", "authors": [ "Marina Cazzola", "Louis Gogniat", "Pablo Spiga" ], "comment": "21 pages", "categories": [ "math.CO", "math.GR" ], "abstract": "Given a permutation group $G$, the derangement graph of $G$ is defined with vertex set $G$, where two elements $x$ and $y$ are adjacent if and only if $xy^{-1}$ is a derangement. We establish that, if $G$ is transitive with degree exceeding 30, then the derangement graph of $G$ contains a complete subgraph with four vertices. As a consequence, if $G$ is a normal subgroup of $A$ such that $|A : G| = 3$, and if $U$ is a subgroup of $G$ satisfying $G = \\bigcup_{a \\in A} U^a$, then $|G : U| \\leq 10$. This result provides support for a conjecture by Neumann and Praeger concerning Kronecker classes.", "revisions": [ { "version": "v1", "updated": "2025-02-03T11:58:56.000Z" } ], "analyses": { "keywords": [ "derangement graph", "praeger concerning kronecker classes", "permutation group", "vertex set", "normal subgroup" ], "note": { "typesetting": "TeX", "pages": 21, "language": "en", "license": "arXiv", "status": "editable" } } }