On progress towards inverse Galois problem over rationals

Question

I have heard that Progress towards the Inverse Galois problem over $\mathbb{Q}$ is very well documented for sporadic groups ($M_{23}$ is the only case open) and for $PSL_n(q)$.

From where I can read more about these results in details. Like papers, articles, books, etc.

Answer 1

All the sporadic groups except for $M_{23}$ and $M_{24}$ are realized over $\mathbb{Q}$ using the rigidity criterion. This technique is explained in all three main textbooks on inverse Galois theory:

• Jean-Pierre Serre - Topics in Galois theory
• Helmut Volklein - Groups as Galois Groups
• Gunter Malle & B. H. Matzat - Inverse Galois theory

For example Volklein explains some of the details applied to the case of $M_{12}$ and the Monster group $M$, while Malle & Matzat present simplified proofs of much more sporadic groups. In the case of the the Monster, you might want to look at Thompson's classic paper:

• John G. Thompson - "Some finite groups which appear as $\mathrm{Gal} (L/K)$, where $K ⊆ \mathbb{Q}(μ_n)$" (1984)

There's also this very informative answer here at MO by user Wanderer.

The case of $M_{24}$, which escapes rigidity, is realized in:

• B. H. Matzat - "Rationality Criteria for Galois extensions" (1987)

The progress on $\mathrm{PSL}_n(q)$ is less uniform. Volklein in his book covers his own regularity result in the case of $n$ even and $n \geq q$, using weak rigidity and braiding action. He also looks at the $\mathrm{PSL}_2(q)$ case, for which Serre (chapter 5) is also a good reference.

There's of course plenty of other results and special cases known. Malle-Matzat is by far the more comprehensive and includes most of the references that you might need.