# Special linear group is characteristic in general linear group

From Groupprops

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

## Contents

## Statement

Suppose is a field and is a natural number. Let denote the group of all invertible matrices over , and denote the subgroup comprising matrices of determinant . Then, is a characteristic subgroup of : every automorphism of sends to itself.

## Related facts

### Stronger facts

## Facts used

- Commutator subgroup of general linear group is special linear group: This result holds except when and has two elements.
- Commutator subgroup is characteristic

## Proof

### The case of a field with two elements

In this case, every invertible matrix has determinant , because is the only nonzero number in the field. Thus, . Since every group is a characteristic subgroup of itself, is characteristic in .

### Other cases

The proof follows from facts (1) and (2).