# Groups and matrices

Let $K$ be the additive group of $\mathbb Z\oplus \mathbb Z$. If $A = \left(\begin{array}{cc} a & b\\ c & d \end{array}\right)$ is an $2\times 2$ matrix where $a, b, c, d$, are in $\mathbb Z$, then $HA*=\langle(a,b), (c,d)\rangle$ is the subgroup generated by rows of matrix $A$.

1) Let $A= \left(\begin{array}{cc} 3 & 1\\ 0 & 5 \end{array}\right)$. Show that $K/HA*$ has order 15.

2) Let $B= \left(\begin{array}{cc} 3 & 1\\ 6 & 7 \end{array}\right)$. Show that $K/HB*$ has order 15.

3) Let $C= \left(\begin{array}{cc} 9 & 8\\ 6 & 7 \end{array}\right)$. Show that $K/HC*$ has order 15.

My thinking: I can use something that restricts the quotient group but I am not sure how to do it?

Thank you for helping out.

You should be able to see how to get the second and third fom the first by using row operations. To see why the first gives an isomorphism to $\mathbb Z_3 \oplus \mathbb Z_5$ is trickier. You need to use the fact that $3$ and $5$ are relatively prime to get rid of the $1$ somehow.
The Smith Normal Form of your matrices is the same: $\left(\begin{array}{cc} 1 & 0\\ 0 & 15 \end{array}\right).$ This shows that all three factor groups are isomorphic to $\mathbb Z_{15}$.