Groebner-Shirshov bases for Schreier extensions of groups

TL;DR

This paper uses Groebner-Shirshov bases to characterize Schreier extensions of groups presented by generators and relations, providing an algorithm to identify such extensions and analyzing their structure via a special total order.

Contribution

It introduces a novel application of Groebner-Shirshov bases to characterize and analyze Schreier extensions of groups, including an algorithm and structural insights.

Findings

Characterization of Schreier extensions using Groebner-Shirshov bases

An algorithm for determining when a group is a Schreier extension

Structural description of Schreier extensions by HNN groups

Abstract

In this paper, by using the Groebner-Shirshov bases, we give characterizations of the Schreier extensions of groups when the group is presented by generators and relations. An algorithm to find the conditions of a group to be a Schreier extension is obtained. By introducing a special total order, we obtain the structure of the Schreier extension by an HNN group.