SCIENCE & INNOVATION

GENERATORS AND RELATIONSHIPS IN GENERALIZED M-TRIANGULAR GROUPS OVER AN ASSOCIATE RING. I

31.10.2023 International Scientific Journal "Science and Innovation". Series A. Volume 2 Issue 10

Satarov Zhoomart, Mamaziaeva Elmira, Mambetov Zhoomart, Orunbaeva N.A.

Abstract. The question of representing linear groups (and related constructions) by generating elements and defining relations has always been of interest in general combinatorial group theory. Today, a large amount of magazine and book materials have already accumulated in this direction. New research methods also emerged. One of them is the universal combinatorial transformation method, the essence of which is to transform words of the selected generative alphabet of the group under study to their standard forms. The paper provides a description through generators and defining relations of generalized m-triangular groups defined over an arbitrary non-zero associative ring. Based on this result, combinatorial descriptions of the projective factors of the named groups are also found. The solution to these problems is based on the mentioned transformation method.

Keywords: generators, relations, quasi-multiplication, quasigroup, generalized m-triangular group, standard forms, transformation of letters, completeness of relations, projective factor.