# Invariable generation of classical groups

Elements ${g_1, \dots, g_k}$ in a group ${G}$ invariably generate if they still generate after an adversary replaces them by conjugates. This is a function of conjugacy classes: we could say that conjugacy classes ${\mathcal{C}_1, \dots, \mathcal{C}_k}$ in a group ${G}$ invariably generate if ${\langle g_1, \dots, g_k\rangle = G}$ whenever ${g_i \in \mathcal{C}_i}$ for each ${i}$. This concept was invented by Dixon to quantify expected running time of the most standard algorithm for computing Galois groups: reduce modulo various primes ${p}$, get your Frobenius element ${g_p}$, and then try to infer what your Galois group is from the information that it contains ${g_p}$ (which is defined only up to conjugacy) for each ${p}$. If ${\textup{Gal}(f)}$ is secretly ${G}$, and you somehow know a priori that ${\textup{Gal}(f) \leq G}$, then the number of primes you need on average to prove that ${\textup{Gal}(f) = G}$ is the expected number of elements it takes to invariably generate ${G}$.

For example, if ${G = S_n}$, then we know that four random elements invariably generate with positive probability, while three random elements almost surely (as ${n\to\infty}$) do not invariably generate. Therefore if ${\textup{Gal}(f) = S_n}$ then it typically takes four primes to prove it.

A few days ago Eilidh McKemmie posted a paper on the arxiv which extends this result to finite classical groups: e.g., if ${G}$ is ${\textup{SL}_n(q)}$ then, for large enough constant ${q}$ and ${n\to\infty}$, four random elements invariably generate with positive probability, but three do not. (The bounded-rank case is rather different in character, and I think two elements suffice.) The proof is pretty cool: invariable generation in ${G}$ is related to invariable generation in the Weyl group, which is either ${S_n}$ or ${C_2 \wr S_n}$, and we already understand invariable generation for these groups (using a small trick for the latter).

I believe the restriction to large enough constant ${q}$ is a technical rather than essential problem. Assuming it can be overcome, we will be able to deduce the following rather clean statement: If ${G}$ is a finite simple group then four random elements invariably generate ${G}$ with probability bounded away from zero. Moreover, if the rank of ${G}$ is unbounded then three random elements do not.

# Mathematical handwriting

I was once told in no uncertain terms to stop using the letter $\xi$ in my talks. I disagree. I think it’s a great letter, especially in scrabble.

I just stumbled acrossed this great page by John Kerl, which contains numerous tips for handwriting in mathematics. I think I differ in one or two places, but I couldn’t agree more with taking a moment to think about it. Maths is communication for the most part, starting with one’s own handwritten notes.

On the subject of excellent references about mathematical communication, I also really enjoyed reading The Grammar According to West, which is something of a Strunk and White for maths papers.