On Transitive Ternary Relational Structures of Order a Prime-Squared

Abstract

We determine the full Sylow \(p\)-subgroup of the automorphism group of transitive \(k\)-ary relational structures of order \(p^2\), \(p\) a prime. We then find the full automorphism group of transitive ternary relational structures of order \(p^2\), for those values of \(p\) for which \({A_p}\) is the only doubly-transitive nonabelian simple group of degree \(p\). Finally, we determine optimal necessary and sufficient conditions for two Cayley \(k\)-ary relational structures of order \(p^2\), \(k < p\), to be isomorphic.