Generalized torsion elements in groups

TL;DR

This paper investigates generalized torsion elements in groups, proving their equivalence to torsion elements in certain classes and providing a method to compute their order in finite groups using character tables.

Contribution

It establishes that in nilpotent and FC-groups, generalized torsion elements are torsion, and introduces a way to determine their order in finite groups via character tables.

Findings

Generalized torsion elements are torsion in nilpotent and FC-groups.

A method to compute generalized order in finite groups using character tables.

Enhanced understanding of torsion properties in specific group classes.

Abstract

A group element is called a generalized torsion if a finite product of its conjugates is equal to the identity. We prove that in a nilpotent or FC-group, the generalized torsion elements are all torsion elements. Moreover, we compute the generalized order of an element in a finite group $G$ using its character table.