# Varieties of elements of given order in simple algebraic groups

## TL;DR

This paper investigates the structure and properties of elements of specific orders in simple algebraic groups, determining the dimension of related subvarieties and applying findings to finite simple quotients of triangle groups.

## Contribution

It provides a complete determination of the dimension of subvarieties of elements of given order in simple algebraic groups, extending previous results and applying to finite group quotients.

## Key findings

- Determined the dimension of subvarieties of elements of order dividing u in simple algebraic groups.
- Extended Lawther's results to groups of adjoint type.
- Proved certain finite quasisimple groups are not quotients of specific triangle groups.

## Abstract

Given a positive integer $u$ and a simple algebraic group $G$ defined over an algebraically closed field $K$ of characteristic $p$, we derive properties about the subvariety $G_{[u]}$ of $G$ consisting of elements of $G$ of order dividing $u$. In particular, we determine the dimension of $G_{[u]}$, completing results of Lawther [7] in the special case where $G$ is of adjoint type. We also apply our results to the study of finite simple quotients of triangle groups, giving further insight on a conjecture we proposed in [10] as well as proving that some finite quasisimple groups are not quotients of certain triangle groups.