# Special linear group is fully 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, Fully characteristic subgroup (?)) in a particular group or type of group (namely, General linear group (?)).

## Statement

Let be a field and be a natural number. Let denote the group of invertible matrices over under multiplication and denote the subgroup comprising matrices of determinant one. is a fully characteristic subgroup of : every endomorphism of restricts to an endomorphism of .

## Facts used

- Commutator subgroup of general linear group is special linear group: This is true except in the case where and the field has exactly two elements.
- Commutator subgroup is fully characteristic

## Proof

### For the field with two elements

In this case, every element of has determinant one, so . Since every group is fully characteristic as a subgroup of itself, is a fully characteristic subgroup of .

### For other cases

In all other cases, the result follows from facts (1) and (2).