# Torsion normal generators of the mapping class group of a non-orientable surface

## TL;DR

This paper investigates the conditions under which periodic elements in the mapping class group of a non-orientable surface generate the entire group or its significant subgroups, focusing on torsion elements and involutions.

## Contribution

It establishes that the normal closure of certain periodic elements contains the twist subgroup, and provides criteria for involutions to normally generate the group.

## Key findings

- Normal closure of elements of order > 2 contains the commutator subgroup.
- For genus g ≥ 7, the commutator subgroup equals the twist subgroup.
- Criteria are given for periodic elements to normally generate the mapping class group.

## Abstract

We show that the normal closure of any periodic element of the mapping class group of a non-orientable surface whose order is greater than 2 contains the commutator subgroup, which for $g\geq 7$ is equal to the twist subgroup, and provide necessary and sufficient conditions for the normal closures of involutions to contain the twist subgroup. Finally, we provide a criterion for a periodic element to normally generate $\mathcal{M}(N_g)$ and give examples.