Special linear group over integers

Definition

Let $n$ be a natural number. The special linear group over integers of degree $n$, denoted $SL(n,\mathbb{Z})$, is defined as follows:

• It is the special linear group $SL(n,R)$ where the base ring $R$ is set as equal to the ring of integers $\mathbb{Z}$.
• It is a subgroup of index two in the automorphism group of the free abelian group of rank $n$, namely, the subgroup of orientation-preserving automorphisms. The whole automorphism group is the general linear group over integers $GL(n,\mathbb{Z})$.

Particular cases

• Special linear group:SL(2,Z): This is the case where $n=2$.
• Special linear group:SL(3,Z): This is the case where $n=3$.