Quotient group (matrices)

Question

What group do we obtain when we quotient $\mathrm{GL}_2 (\mathbb Z )$ by $\mathrm{ SL}_2 (\mathbb Z) $ ?

Answer 1

Yeah map $GL_2(\Bbb Z)$ to $\Bbb Z/2\Bbb Z$ by $M\mapsto\det(M)$. The kernel is $SL_2(\Bbb Z)$. So the quotient is $\Bbb Z/2\Bbb Z$.

Answer 2

Hint: Define $\phi: \mathbb{GL}_2(\mathbb{Z}) \to \{1,-1\}$ by sending $A \to \det(A)$. Look at the Kernel of this map and use Fundamental Homomorphism Theorem.