TITLE: Quotient group $\mathbb Z^n/\ \text{im}(A)$
QUESTION [7 upvotes]: Let $A$ be an $n \times n$ matrix with integer coefficients and nonzero determinant. Can we say something about $ \mathbb{Z}^n /\ \text{im}( \phi )$ (here $\phi : v \mapsto Av$ )? 

This problem has arised as I was solving some problem in homology theory.

REPLY [5 votes]: This can be seen from applying Smith's normal form to the matrix $A$ to get an equivalent matrix $$A'={\rm diag}\;(d_1,\ldots,d_n)$$
Then each $d_i\neq 0$ since the determinant is nonzero. Now one can produce distinct linear combinations $\lambda_1 \overline{e_1}+\cdots+\lambda_n\overline{e_n}$ in $d_1d_2\cdots d_n=\det A$ ways, and one can show these are all, which gives Eric's result.