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Abstract
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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.
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