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KRONECKER CLASSES AND CLIQUES IN DERANGEMENT GRAPHS
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- Journal:
- Journal of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 13 April 2026, pp. 1-29
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- Article
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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. In the process, we determine all transitive groups whose derangement graph does not contain a complete subgraph on four vertices. As a consequence, if G is a normal subgroup of A such that 
$|A : G| = 3$ and U is a subgroup of G satisfying 
$G = \bigcup _{a \in A} U^a$, then 
$|G : U| \leq 10$. This provides support for a conjecture by Neumann and Praeger concerning Kronecker classes.
