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By $\text{GL}^+(n,\mathbb{Z})$ we mean the set of $n×n$ invertible matrices with positive determinant and entries from $\mathbb{Z}$. For given $A \in \text{GL}^+(n,\mathbb{Z})$ let $F_A=\mathbb{Z}^n/A\mathbb{Z}^n$ denote the cokernel of $A$. I would like to compute $|F_A|$. How does one do this?

Shawn O'Hare
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    What do you mean by $;A\Bbb Z^n;$ ? – Timbuc Jul 18 '14 at 20:06
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    @Timbuc Most likely the image of $A$. – Shawn O'Hare Jul 18 '14 at 20:17
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    @Timbuc $A\mathbb{Z}^n={Av\mid v\in \mathbb{Z}^n}$, $v$ here considered as column vector –  Jul 18 '14 at 20:33
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    Q: How does one do this? A: Smith normal form. – anon Jul 18 '14 at 20:38
  • @blue If B is the Smith normal form of A, do you mean that $F_A=F_B$ –  Jul 18 '14 at 20:45
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    Yes. Well, $F_A\cong F_B$ technically. – anon Jul 18 '14 at 20:47
  • @blue Not sure how to prove $F_A\cong F_B$. I guess that we need to prove that $F_A=F_{GA}=F_{AG}$ for every $G\in \text{GL}^+(n,\mathbb{Z})$ with $\det(G)=1$? :) –  Jul 18 '14 at 21:14
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    We have $\Bbb Z^n/AG\Bbb Z^n=\Bbb Z^n/A\Bbb Z^n$ for every matrix $G$ such that $G\Bbb Z^n=\Bbb Z^n$ (namely, the invertible matrices), and we have $G:\Bbb Z^n/A\Bbb Z^n\cong\Bbb Z^n/GA\Bbb Z^n$ via $v+A\Bbb Z^n\mapsto Gv+GA\Bbb Z^n$. – anon Jul 18 '14 at 21:19

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