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 defining ingredient::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 $$n = 3$$.