An S-ring (a Schur ring) is said to be separable with respect to a class of groups 
if every its algebraic isomorphism to an S-ring over a group from 
is induced by a combinatorial isomorphism. It is proved that every Schur ring over an Abelian group G of order 4p, where p is a prime, is separable with respect to the class of Abelian groups. This implies that the Weisfeiler-Lehman dimension of the class of Cayley graphs over G is at most 3.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Evdokimov, “Schurity and separability of association schemes,” Habilitationschrift St. Peterburg State Univ. (2004).
S. Evdokimov and I. Ponomarenko, “On a family of Schur rings over a finite cyclic group,” Algebra Analiz, 13, No. 3, 139–154 (2001).
S. Evdokimov and I. Ponomarenko, “Permutation group approach to association schemes,” European. J. Combin., 30, 1456–1476 (2009).
S. Evdokimov and I. Ponomarenko, “Schurity of S-rings over a cyclic group and generalized wreath product of permutation groups,” Algebra Analiz, 24, No. 3, 84–127 (2012).
S. Evdokimov and I. Ponomarenko, “Schur rings over a product of Galois rings,” Beitr. Algebra Geom., 55, No. 1, 105–138 (2014).
S. Evdokimov and I. Ponomarenko, “On separability problem for circulant S-rings,” Algebra Analiz, 28, No. 1, 32–51 (2016).
S. Evdokimov, I. Kovács, and I. Ponomarenko, “On schurity of finite Abelian groups,” Commun. Algebra, 44, No. 1, 101–117 (2016).
S. Kiefer, I. Ponomarenko, and P. Schweitzer, “TheWeisfeiler-Lehman dimension of planar graphs is at most 3,” in: 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), IEEE, New Jersey (2017).
M. Klin, C. Pech, and S. Reichard, “COCO2P – a GAP package, 0.14,” http://www.math.tu-dresden.de/ pech/COCO2P.
M. Muzychuk and I. Ponomarenko, “Schur rings,” European J. Combin., 30, 1526–1539 (2009).
R. Nedela and I. Ponomarenko, “Recognizing and testing isomorphism of Cayley graphs over an abelian group of order 4p in polynomial time,” arXiv:1706.06145 [math.CO], 1–22 (2017).
G. Ryabov, “On separability of Schur rings over abelian p-groups,” Algebra and Logic, 57, No. 1, 73–101 (2018).
I. Schur, “Zur theorie der einfach transitiven Permutationgruppen,” S.-B. Preus Akad. Wiss. Phys.-Math. Kl., 18, No. 20, 598–623 (1933).
H. Wielandt, Finite Permutation Groups, Academic Press, New York–London (1964).
B. Weisfeiler and A. Lehman, “Reduction of a graph to a canonical form and an algebra which appears in the process,” NTI, 2, No. 9, 12–16 (1968).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 470, 2018, pp. 179–193.
Rights and permissions
About this article
Cite this article
Ryabov, G. Eparability of Schur Rings Over an Abelian Group of Order 4p. J Math Sci 243, 624–632 (2019). https://doi.org/10.1007/s10958-019-04563-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-019-04563-9