Presentation of commutative group by integer matrix

Question

The task states: "Find a integer matrix $A \in Mat_{3 \times 4} \mathbb Z$ which presents commutative group $\mathbb Z \times \mathbb Z_2 \times \mathbb Z_3 \times \mathbb Z_4$ and find it's Smith normal form". My thought is that SMT of such matrix should be $$ \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 4 \\ \end{pmatrix} $$ but I have trouble finding matrix itself. Am I right about SMT form? How do I find such a matrix?

Answer 1

Your matrix below is fine for $4$ generators $$\begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 4 \\ \end{pmatrix} $$ but it's not in Smith normal form because we don't have $2 \mid 3 \mid 4$.

Since $\mathbb Z \times \mathbb Z_2 \times \mathbb Z_3 \times \mathbb Z_4 \cong \mathbb Z \times \mathbb Z_2 \times \mathbb Z_{12}$, the Smith normal form will be $$ \begin{pmatrix} 0 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 12 \\ \end{pmatrix} $$ Note that we now need only $3$ generators.