query: How to show that $\#(\mathbb Z^n/N) = \det(A)$ where $N$ is the subgroup of $\mathbb Z^n$ generated by columns of a square matrix $A$
positive_passages.docid: 137422
positive_passages.text: $\mathrm{card}(\mathbb{Z}^n/M\mathbb{Z}^n) = |\det(M)|$?
negative_passages.docid: 98507
negative_passages.text: $\text{Aut}(F)$ is isomorphic to the multiplicative group of all $n\times n$ matrices over $\mathbb Z$
negative_passages.docid: 149578
negative_passages.text: Subgroups of $(\mathbb Z_n,+)$
negative_passages.docid: 39895
negative_passages.text: Showing a free abelian group is generated by its basis
negative_passages.docid: 152637
negative_passages.text: Determinant of a specific circulant matrix, $A_n$
negative_passages.docid: 116936
negative_passages.text: Commutator subgroup and subgroup generated by square.
negative_passages.docid: 133449
negative_passages.text: A proof that an orthogonal matrix with a determinant 1 is a rotation matrix
negative_passages.docid: 11003
negative_passages.text: Proving that $G/N$ is an abelian group
negative_passages.docid: 37252
negative_passages.text: How can I show that $\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}^n = \begin{pmatrix} 1 & n \\ 0 & 1\end{pmatrix}$?
negative_passages.docid: 93169
negative_passages.text: Show that $(\mathbb Z[x],+)$ and $(\mathbb Q_{>0},\cdot)$ are isomorphic groups
negative_passages.docid: 75068
negative_passages.text: subgroups of $\mathbb Z / 2 \mathbb Z \times \mathbb Z / 2 \mathbb Z$
negative_passages.docid: 69641
negative_passages.text: Determinant identity: $\det M \det N = \det M_{ii} \det M_{jj} - \det M_{ij}\det M_{ji}$
negative_passages.docid: 64742
negative_passages.text: Prove a residue matrix $A$ (with coefficients in $\mathbb Z_n)$ has an inverse if and only if $\gcd(\det A,n) = 1$
negative_passages.docid: 137739
negative_passages.text: Prove that the group isomorphism $\mathbb{Z}^m \cong \mathbb{Z}^n$ implies that $m = n$
negative_passages.docid: 84522
negative_passages.text: A better proof for $\det(P) = \pm1$ if $P$ is an orthogonal matrix
negative_passages.docid: 81181
negative_passages.text: Prove that $\det(M-I)=0$ if $\det(M)=1$ and $MM^T=I$
negative_passages.docid: 131212
negative_passages.text: Prove that if $A$ is regular then $\operatorname{adj}(\operatorname{adj}(A)) = (\det A)^{n-2} A$