Braid orbits and the Mathieu group $M_{23}$ as Galois group

TL;DR

This paper reviews the current efforts and challenges in realizing the Mathieu group M_{23} as a Galois group over the rationals, highlighting new braid invariants and their implications for the inverse Galois problem.

Contribution

It introduces new braid invariants and discusses their potential in constructing Galois realizations, advancing understanding of the inverse Galois problem for M_{23}.

Findings

New braid invariants discovered beyond Fried's lifting invariant.

Braid orbit computations suggest pathways for Galois realizations of M_{24}.

Realization of M_{23} over  remains unresolved.

Abstract

At present, the inverse Galois problem over $\mathbb{Q}$ is unsolved for the Mathieu group $M_{23}$. Here an overview of the current state in realizing $M_{23}$ as Galois group using the rigidity method and the action of braids is given. Computing braid orbits for $M_{23}$ revealed new invariants of the action of braids in addition to Fried's lifting invariant. These invariants can be used to construct generic braid orbits and more Galois realizations over $\mathbb{Q}$ for the Mathieu group $M_{24}$, but until now did not lead to success for realising $M_{23}$ as Galois group over $\mathbb{Q}$. Thus $M_{23}/\mathbb{Q}$ remains open. Finally, heuristics for searching suitable class vectors with regard to the realization of groups as Galois groups are given.