## Abstract

For any finite group G, and any positive integer n, we construct an association scheme which admits the diagonal group D

A transitive permutation group G is said to be

We construct another association scheme, finer than the above scheme if n>3, from the Latin hypercube consisting of n-tuples of elements of G with product the identity.

_{n}(G) as a group of automorphisms. The rank of the association scheme is the number of partitions of n into at most |G| parts, so is p(n) if |G| ≥ n; its parameters depend only on n and |G|. For n=2, the association scheme is trivial, while for n=3 its relations are the Latin square graph associated with the Cayley table of G and its complement.A transitive permutation group G is said to be

*AS-free*if there is no non-trivial association scheme admitting G as a group of automorphisms. A consequence of our construction is that an AS-free group must be either 2-homogeneous or almost simple.We construct another association scheme, finer than the above scheme if n>3, from the Latin hypercube consisting of n-tuples of elements of G with product the identity.

Original language | English |
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Pages (from-to) | 357-364 |

Journal | Australasian Journal of Combinatorics |

Volume | 75 |

Issue number | 3 |

Publication status | Published - 27 Oct 2019 |

## Keywords

- Association scheme
- Diagonal group
- Latin square