# Finite order elements in the integral symplectic group

## TL;DR

This paper investigates the orders of elements in the integral symplectic group, proving that the set of possible orders is bounded for each dimension and demonstrating that the number and maximum order grow at least exponentially with the dimension.

## Contribution

It establishes the boundedness of element orders in symplectic groups and analyzes the exponential growth of their count and maximum order as the dimension increases.

## Key findings

- Set of element orders is bounded for each g
- Number of possible orders grows at least exponentially with g
- Maximum order of elements also grows at least exponentially

## Abstract

For $g\in \mathbb{N}$, let $G=\Sp(2g,\mathbb{Z})$ be the integral symplectic group and $S(g)$ be the set of all positive integers which can occur as the order of an element in $G$. In this paper, we show that $S(g)$ is a bounded subset of $\mathbb{R}$ for all positive integers $g$. We also study the growth of the functions $f(g)=|S(g)|$, and $h(g)=max\{m\in \mathbb{N}\mid m\in S(g)\}$ and show that they have at least exponential growth.