# Projective special linear group is simple

This article gives the statement, and possibly proof, of a particular group or type of group (namely, Projective special linear group (?)) satisfying a particular group property (namely, Simple group (?)).

## Statement

Let $k$ be a field and $n$ be a natural number greater than $1$. Then, the projective special linear group $PSL_n(k)$ is a simple group provided one of these conditions holds:

• $n \ge 3$.
• $k$ has at least four elements.

## Facts used

1. Special linear group is perfect: Under the same conditions ($n \ge 3$ or $k$ has at least four elements), the special linear group $SL_n(k)$ is a perfect group: it equals its own derived subgroup.
2. Perfectness is quotient-closed: The quotient of a perfect group by a normal subgroup is perfect.
3. Abelian normal subgroup of core-free maximal subgroup is contranormal implies derived subgroup of whole group is monolith

## Proof

The proof proceeds in the following steps:

1. $PSL_n(k)$ satisfies the hypotheses for fact (3): Consider the natural action of $PSL_n(k)$ on the projective space $\mathbb{P}^{n-1}(k)$. This is a primitive group action, and the stabilizer of any point is thus a core-free maximal subgroup. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
2. The commutator subgroup of $PSL_n(k)$ is contained in every nontrivial normal subgroup of $PSL_n(k)$: This follows from the previous step and fact (3).
3. $PSL_n(k)$ equals its own commutator subgroup when $n \ge 3$ or $k$ has at least four elements: This follows from facts (1) and (2).
4. $PSL_n(k)$ is simple when $n \ge 3$ or $k$ has at least four elements: : This follows from the last two steps.