Covering groups of $M_{22}$ as regular Galois groups over $\mathbb{Q}$

TL;DR

This paper proves that the full covering groups of the Mathieu group M22 and its automorphism group can be realized as Galois groups over the rational numbers, filling a gap in the existing literature.

Contribution

It demonstrates the realization of the full covering groups of M22 and Aut(M22) as Galois groups over Q, which was previously unresolved.

Findings

Full covering groups of M22 and Aut(M22) occur as Galois groups over Q.

Fills a gap in the classification of Galois groups related to Mathieu groups.

Provides explicit constructions of such Galois extensions.

Abstract

We close a gap in the literature by showing that the full covering groups of the Mathieu group $M_{22}$ and of its automorphism group $Aut(M_{22})$ occur as the Galois group of $\mathbb{Q}$-regular Galois extensions of $\mathbb{Q}(t)$.