Elementary proof that Zₚ⁴ is a DCI-group

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A finite group R is a DCI-group if, whenever S and T are subsets of R with the Cayley graphs Cay(R,S) and Cay(R,T) isomorphic, there exists an automorphism φ of R with Sφ=T. Elementary abelian groups of order p⁴ or smaller are known to be DCI-groups, while those of sufficiently large rank are known not to be DCI-groups. The only published proof that elementary abelian groups of order p⁴ are DCI-groups uses Schur rings and does not work for p=2 (which has been separately proven using computers). This paper provides a simpler proof that works for all primes. Some of the results in this paper also apply to elementary abelian groups of higher rank, so may be useful for completing our determination of which elementary abelian groups are DCI-groups.