Let $G=\left\{\begin{bmatrix}a & b \\ c & d\end{bmatrix}:a,b,c,d\in\mathbb{Z}\right\}$ be the group under matrix addition and $H$ be the subgroup of $G$ consisting of matrices with even entries. Find the order of the quotient group $G/H$.
How should I solve this problem?
$G\simeq \mathbb Z^4$ (the isomorphism is given by $\begin{bmatrix}a & b \\ c & d\end{bmatrix}\mapsto (a,b,c,d)$) and $H\simeq (2\mathbb Z)^4$ $\Rightarrow$ $G/H\simeq ({\mathbb Z}/2\mathbb Z)^4$ and this shows that $|G/H|=16$.
First way: Try to prove that the set
$$\left\{\;\begin{pmatrix}\alpha&\beta\\\gamma&\delta\end{pmatrix}\;\;;\;\;\alpha,\beta,\gamma,\delta\in\{0,1\}\;\right\}$$
is a complete set of representatives of the different cosets in $\,G/H\,$ (this implies that you also must prove these represent all the different elements in the quotient group). How many are there?
Second way: Define
$$\,\phi: G\to K:=C_2\times C_2\times C_2\times C_2\,\,,\,\,\phi\begin{pmatrix}a&b\\c&d \end{pmatrix}:=\left(a,b,c,d\right)\pmod 2\in K\,$$
Check the above is a group homomorphism...what is its kernel?
