Special linear group is characteristic in general linear group

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 (?)).

Statement

Suppose $k$ is a field and $n$ is a natural number. Let $GL_{n}(k)$ denote the group of all invertible $n\times n$ matrices over $k$, and $SL_{n}(k)$ denote the subgroup comprising matrices of determinant $1$. Then, $Sl_{n}(k)$ is a characteristic subgroup of $GL_{n}(k)$: every automorphism of $GL_{n}(k)$ sends $SL_{n}(k)$ to itself.

Related facts

Stronger facts

• Special linear group is fully characteristic in general linear group

Facts used

1. Commutator subgroup of general linear group is special linear group: This result holds except when $n=2$ and $k$ has two elements.
2. Commutator subgroup is characteristic

Proof

The case of a field with two elements

In this case, every invertible matrix has determinant $1$, because $1$ is the only nonzero number in the field. Thus, $SL_{n}(k)=GL_{n}(k)$. Since every group is a characteristic subgroup of itself, $SL_{n}(k)$ is characteristic in $GL_{n}(k)$.

Other cases

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