  Let $G$ be a group acting faithfully and transitively on $\Omega_i$ for
$i=1,2$. A famous theorem by Burnside implies the following fact: If
$|\Omega_1|=|\Omega_2|$ is a prime and the rank of one of the actions is
greater than two, then the actions are equivalent, or equivalently
$|(\alpha,\beta)^G|=|\Omega_1|=|\Omega_2|$ for some $(\alpha,\beta)\in
\Omega_1\times \Omega_2$.
  In this paper we consider a combinatorial analogue to this fact through the
theory of coherent configurations, and give some arithmetic sufficient
conditions for a coherent configuration with two homogeneous components of
prime order to be uniquely determined by one of the homogeneous components.
