Matricial Rrepresentations of Certain Finitely Presented Groups Generated By Order-2 Generator and their Applications

TL;DR

This paper explores matrix representations of specific finitely presented groups with order-2 generators, linking group algebra inverses to matrix properties and revealing connections to Lucas numbers.

Contribution

It introduces a novel matrix representation framework for these groups and characterizes element inverses using matrices, uncovering links to Lucas numbers.

Findings

Inverses of group elements are characterized via matrices in the group algebra.

A relationship between the trace of the radial operator and Lucas numbers is established.

The study provides new insights into the algebraic structure of these groups through matrix analysis.

Abstract

In this paper, we study matricial representations of certain finitely presented groups with N-generators of order-2. As an application, we consider a group algebra under our representations. Specifically, we characterize the inverses of all group elements in terms of matrices in the group algebra. From the study of this characterization, we realize there are close relations between the trace of the radial operator and the Lucas numbers appearing in the Lucas triangle.