Enumeration of Schur rings over small groups

Abstract

By optimizing the algorithms used in COCO and COCO-II, we enumerated all Schur rings over the groups of orders up to 63. A few statistical views of results with respect to Schur property, amount and type of generators and primitivity are presented. Discussion of the details of the old algorithms and the improvements we implemented in order to achieve those results is included. We compare the results to similar computerized efforts (Hanaki and Miyamoto, Pech and Reichard, Heinze), as well as to theoretical classifications of Schur groups. The computer based results may assist the theoretical efforts to classify all Schur groups, over abelian and non-abelian groups. © 2014 Springer International Publishing.