General linear group over integers

Definition

Let $n$ be a natural number. The general linear group of order $n$ over integers, denoted $GL(n,\mathbb {Z} )$ or $GL_{n}(\mathbb {Z} )$, is defined in the following equivalent ways:

1. It is the general linear group of order $n$ over the ring $\mathbb {Z}$, defined as the ring of integers.
2. It is the automorphism group of the free abelian group of rank $n$, i.e., the group $\mathbb {Z} ^{n}$.