A note on invariable generation of nonsolvable permutation groups

TL;DR

This paper investigates the probability that two random permutations in the symmetric group generate a nonsolvable subgroup, with applications to Galois groups of random polynomials and their solvability.

Contribution

It establishes an asymptotic result on the likelihood of random permutations invariably generating nonsolvable subgroups, linking permutation properties to Galois theory.

Findings

High probability of random pairs generating nonsolvable groups

Application to Galois groups of random polynomials

Reduction modulo two primes often suffices to prove nonsolvability

Abstract

We prove a result on the asymptotic proportion of randomly chosen pairs of permutations in the symmetric group $S_n$ which "invariably" generate a nonsolvable subgroup, i.e., whose cycle structures cannot possibly both occur in the same solvable subgroup of $S_n$. As an application, we obtain that for a large degree "random" integer polynomial $f$, reduction modulo two different primes can be expected to suffice to prove the nonsolvability of $Gal(f/\mathbb{Q})$.