Generalized torsion orders of generalized torsion elements

TL;DR

This paper investigates the properties of generalized torsion elements in groups, focusing on the minimal number of conjugates needed to produce the identity, and introduces restrictions based on algebraic invariants.

Contribution

It introduces new restrictions on generalized torsion orders using $G$-invariant norms and Alexander polynomials, advancing understanding of group element behavior.

Findings

Restrictions on generalized torsion orders established

Use of $G$-invariant norms to analyze torsion elements

Application of Alexander polynomials to derive new constraints

Abstract

A non-trivial element of a group is a generalized torsion element if some products of its conjugates is the identity. The minimum number of such conjugates is called a generalized torsion order. We provide several restrictions for generalized torsion orders by using $G$-invariant norm and Alexander polynomials.