Abstract
An S-ring (a Schur ring) is said to be separable with respect to a class of groups \({\mathcal {K}}\) if every algebraic isomorphism from the S-ring in question to an S-ring over a group from \({\mathcal {K}}\) is induced by a combinatorial isomorphism. A finite group G is said to be separable with respect to \({\mathcal {K}}\) if every S-ring over G is separable with respect to \({\mathcal {K}}\). We prove that every abelian group G of order 9p, where p is a prime, is separable with respect to the class of all finite abelian groups. Modulo previously obtained results, this completes a classification of noncyclic abelian groups of odd order that are separable with respect to the class of all finite abelian groups. This also implies that the relative Weisfeiler–Leman dimension of a Cayley graph over G with respect to the class of all Cayley graphs over abelian groups is at most 2.
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The work is supported by the Russian Foundation for Basic Research (project 18-31-00051).
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Ryabov, G. Separability of Schur Rings Over Abelian Groups of Odd Order. Graphs and Combinatorics 36, 1891–1911 (2020). https://doi.org/10.1007/s00373-020-02206-4
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DOI: https://doi.org/10.1007/s00373-020-02206-4