Special linear group is characteristic in 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.

Stronger facts

 * Special linear group is fully characteristic in general linear group

Facts used

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

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).