# Special linear group is perfect

From Groupprops

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

## Contents

## Statement

Let be any field. Consider the special linear group : the group of matrices over that have determinant . Then, the following are true:

- For , is a perfect group for any field .
- For , is a perfect group if has more than three elements.

## Related facts

- Derived subgroup of general linear group is special linear group
- Symplectic group is perfect
- Derived subgroup of orthogonal group is special orthogonal group

## Facts used

- Elementary matrices generate the special linear group
- Every elementary matrix is a commutator of unimodular matrices: If or and has more than three elements, every elementary matrix is a commutator of two elements of .

## Proof

The proof follows by piecing together facts (1) and (2).