# Projective special linear group is simple

From Groupprops

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 (?)).

## Contents

## Statement

Let be a field and be a natural number greater than . Then, the projective special linear group is a simple group provided one of these conditions holds:

- .
- has at least four elements.

## Facts used

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

## Related facts

### Related facts about special linear group and projective special linear group

- Special linear group is perfect
- Special linear group is quasisimple
- Projective special linear group equals alternating group in only finitely many cases

### Related facts about simplicity of linear groups

- Projective symplectic group is simple
- Projective special orthogonal group for bilinear form of positive Witt index is simple
- projective special orthogonal group over reals is simple

## Proof

The proof proceeds in the following steps:

- satisfies the hypotheses for fact (3): Consider the natural action of on the projective space . 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] - The commutator subgroup of is contained in every nontrivial normal subgroup of : This follows from the previous step and fact (3).
- equals its own commutator subgroup when or has at least four elements: This follows from facts (1) and (2).
- is simple when or has at least four elements: : This follows from the last two steps.