Right Splitting, Galois Correspondence, Galois Representations and Inverse Galois Problem
Chandrasheel Bhagwat, Shubham Jaiswal
TL;DR
The paper advances the inverse Galois problem by developing a cohesive framework based on right splitting of exact sequences, the Galois correspondence, and algebraic operations on Galois representations to realize new semidirect product groups over $\mathbb{Q}$. It establishes general theorems showing that $(H_1\times\cdots\times H_l)\rtimes G_1/H_1$ is realizable under suitable hypotheses and applies these results to constructions involving $GL_2(\mathbb{F}_p)$, $SL_2(\mathbb{F}_p)$, and $PSL_2(\mathbb{F}_p)$, yielding explicit realizations such as $(\mathrm{SL}_2(\mathbb{F}_p)\times \mathrm{SL}_2(\mathbb{F}_p))\rtimes \mathbb{Z}/(p-1)\mathbb{Z}$ and $(\mathrm{PSL}_2(\mathbb{F}_p)\times \mathrm{PSL}_2(\mathbb{F}_p))\rtimes \mathbb{Z}/2\mathbb{Z}$ for primes $p\ge 5$. Leveraging induced representations and modular forms, it shows how $\mathrm{PSL}_2(\mathbb{F}_p)\rtimes \mathbb{Z}/2\mathbb{Z}$ and related groups arise as Galois groups over $\mathbb{Q}$, and demonstrates that direct sums and tensor products of Galois representations preserve realizability, enabling construction of larger groups from simpler components. The work thus broadens the catalog of explicitly realizable finite groups over $\mathbb{Q}$ and provides a versatile toolkit for generating new instances via Galois representations and composition methods.
Abstract
In this article, we realize some groups as Galois groups over rational numbers and finite extension of rational numbers by studying right splitting of some exact sequences, Galois correspondence and algebraic operations on Galois representations.