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Elementary proof that \(\mathbb{Z}_p^4\) is a DCI-group. (English) Zbl 1310.05116
Summary: A finite group \(R\) is a \(\mathrm{DCI}\)-group if, whenever \(S\) and \(T\) are subsets of \(R\) with the Cayley graphs \(\mathrm{Cay}(R, S)\) and \(\mathrm{Cay}(R, T)\) isomorphic, there exists an automorphism \(\varphi\) of \(R\) with \(S^\varphi = T\).
Elementary abelian groups of order \(p^4\) or smaller are known to be \(\mathrm{DCI}\)-groups, while those of sufficiently large rank are known not to be \(\mathrm{DCI}\)-groups. The only published proof that elementary abelian groups of order \(p^4\) are \(\mathrm{DCI}\)-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 \(\mathrm{DCI}\)-groups.

MSC:
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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