Separability of Schur Rings Over Abelian Groups of Odd Order
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ORIGINAL PAPER
Separability of Schur Rings Over Abelian Groups of Odd Order Grigory Ryabov1,2 Received: 18 December 2019 / Revised: 7 June 2020 Ó Springer Japan KK, part of Springer Nature 2020
Abstract An S-ring (a Schur ring) is said to be separable with respect to a class of groups K if every algebraic isomorphism from the S-ring in question to an S-ring over a group from K is induced by a combinatorial isomorphism. A finite group G is said to be separable with respect to K if every S-ring over G is separable with respect to 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. Keywords Schur rings Cayley graphs Cayley graph isomorphism problem
Mathematics Subject Classification 05E30 05C60 20B35
1 Introduction A Schur ring or S-ring over a finite group G can be defined as a subring of the group ring ZG that is a free Z-module spanned by a partition of G closed under taking inverse and containing the identity element e of G as a class (see Sect. 2 for the exact definition). The elements of the partition are called the basic sets of the S-ring. The work is supported by the Russian Foundation for Basic Research (project 18-31-00051). & Grigory Ryabov [email protected] 1
Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, 630090 Novosibirsk, Russia
2
Novosibirsk State University, 1 Pirogova st., 630090 Novosibirsk, Russia
123
Graphs and Combinatorics
The first construction of such ring was proposed by Schur [24]. The general theory of S-rings was developed by Wielandt in [26]. Schur and Wielandt used S-rings to study permutation groups containing regular subgroups. Concerning the theory of Srings, we refer the reader to [1, 16]. Let A and A0 be S-rings over groups G and G0 respectively. A (combinatorial) isomorphism from A to A0 is defined to be a bijection f : G ! G0 satisfying the following condition: for every basic set X of A there exists a basic set X 0 of A0 such that f is an isomorphism of the Cayley graphs Cay ðG; XÞ and Cay ðG0 ; X 0 Þ. An algebraic isomorphism from A to A0 is defined to be a bijection from the set of basic sets of A to the set of basic sets of A0 that preserves the structure constants (with respect to the standard bases of A and A0 ). Every algebraic isomorphism is extended by linearity to the ring isomorphism from A to A0 . One can verify that every combinatorial isomorphism induces the algebraic one. However, the converse statement is not true in general (see [3]). Let K be a class of groups. Following [5], we say that an S-ring A is separable with respect to K if every algebraic isom
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