# 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.

## 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).