
> 
> Let $GL\_{\eta}(n,\mathbb{Z})=\left\{a\in
>  GL(n,\mathbb{R})\cap M^{n\times n}(\mathbb{Z})|det(a)=\eta\right\}$. Prove that there exists a
>  finite number of matrices $a\_i$ in $GL\_{\eta}(n,\mathbb{Z})$ such that
>  any matrix $a\in GL\_{\eta}(n,\mathbb{Z})$ can be written as
>  $a=a\_i\alpha$, where $\alpha \in SL(n,\mathbb{Z})$
> 
> 
> 


Could someone give me some ideas or references for that fact? Thank you so much!



Decomposition of integral non-generate matrices
The magic words are *Hermite Normal Form*. If you read the [Wikipedia article](https://en.wikipedia.org/wiki/Hermite_normal_form), you will discover that the magic matrix $H$ is upper-triangular, with every off-diagonal element *smaller* than the (maximum, for the sake of argument) diagonal element. Since the product of the diagonal elements is equal to $\eta,$ their maximum is at most $\eta,$ and that means that there are finitely many (at most something like $(2\eta)^{n(n+1)/2}$) possibilities.


