Problem Set 06

Read

The overarching goal is to learn about the groups $\operatorname{PSL}_n(F)$ with a focus on $\operatorname{PSL}_2(F)$ for $F$ a finite field and its simplicity (usually).

1. Skim pages 1–5 of Conrad’s notes on the Simplicity of $\operatorname{PSL}_n(F)$. It begins by introducing some important permutation group-theoretic ideas like double-transitivity and a version of Iwasawa’s Lemma. As a first application of Iwasawa’s Lemma, Conrad gives a quick proof of the simplicity of $\operatorname{Alt}(5)$ in Example 2.11. Then Conrad works towards a proof that $\operatorname{PSL}_2(F)$ is simple, which is achieved in Theorem 3.5.
2. If you want to take a deeper dive into projective groups and projective geometries, take a look at Sections 1.4–2.5 of Cameron’s Notes on Classical Groups.

To Work On

1. Write up—in your own words—a proof (or outline) of the simplicity of $\operatorname{PSL}_2(F)$ when $|F|\ge 4$.

2. For $\mathbb{F}_{q}$ a field of order $q$, work out a formula in terms of $q$ for the order of $\operatorname{PSL}_2(\mathbb{F}_{q})$, and use this to show that $|\operatorname{PSL}_2(\mathbb{F}_{7})| = 168$.

3. More to come…