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Let $G$ be the group of all $2\times 2$ matrices with integer entries and let $H$ be the subgroup of all $2\times 2$ matrices with even integer entries. Then I want to find the index of $H$ in $G$.

I think it is $16$. As all cosets will be of the form $\begin{pmatrix}a&b \\ c& d\end{pmatrix}+H$. Now we can choose $16$ such matrices. Am I right?

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  • $\begingroup$ why $16$?? you might want to share your thoughts... What do you say? $\endgroup$ – user87543 Feb 14 '14 at 16:27
  • $\begingroup$ I think each place in the matrix can be chosen by 0 or 1. But I am not sure. $\endgroup$ – Anupam Feb 14 '14 at 16:28
  • $\begingroup$ Since you have a "group of matrices with integer entries", would your group operation be addition? $\endgroup$ – Arthur Feb 14 '14 at 16:31
  • $\begingroup$ yes, $G$ is an additive group $\endgroup$ – Anupam Feb 14 '14 at 16:32
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The exact sequence $0\to 2\mathbb Z\to\mathbb Z\to\mathbb Z\diagup 2\mathbb Z\to 0$ gives rise to the exact sequence: $$0\to\text{Mat}(2,2\mathbb Z)\to\text{Mat}(2,\mathbb Z)\to\text{Mat}(2,\mathbb Z\diagup 2\mathbb Z)\to 0.$$

Thus: $$|\text{Mat}(2,\mathbb Z):\text{Mat}(2,2\mathbb Z)|=|\text{Mat}(2,\mathbb Z\diagup 2\mathbb Z)|=|\mathbb Z\diagup 2\mathbb Z|^4=16.$$

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Yes, you're right. We can choose these representativesof cosets:$\left(\begin{array}{cccc} a& b\\c& d\end{array}\right), a,b,c,d\in\{0,1\}$. Then each element $g\in G$ is equal to $\left(\begin{array}{cccc} a& b\\c& d\end{array}\right)+h$ for some element $h\in H$, there entrie of $\left(\begin{array}{cccc} a& b\\c& d\end{array}\right)$ is $0$ if corresponding entrie in $g$ is even and odd, otherwise.

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