Classification of highly symmetrical translation loops of order $$2p, p$$ prime

Abstract

Using tools from algebraic graph theory, we prove that for every odd prime \(p\) there is just one proper loop of order \(2p\) such that the automorphism group of the corresponding rank 5 association scheme contains a regular subgroup of order \(4p^2\). Although our results were initially obtained on the basis of theoretical generalization of a tremendous number of computer algebra experiments, our final computer-free proof uses quite elementary arguments from group theory and combinatorics. In this paper we provide this computer-free proof, as well as discuss further suggested use of our methods.