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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?

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Before finding the order, can you explicitly write down the elements of $G/H$? –  user1551 Dec 30 '12 at 11:08

2 Answers 2

$G\simeq \mathbb Z^4$ (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 $|G/H|=16$.

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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?

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