The set $G=\left\{\begin{pmatrix}1 & n \\
0 & 1 \\
\end{pmatrix}\mid n\in \Bbb Z\right\}$ with the operation of matrix multiplication is a group. Show that $$\phi:\Bbb Z \to G,$$ $$\phi(n)=\begin{pmatrix}1 & n \\
0 & 1 \\
\end{pmatrix}$$
is a group isomorphism (where the operation on $\Bbb Z$ is ordinary addtion).
TO show it's isomorphism: I know I must show one-to-one, onto and homomorphism. I've done these examples before but never with matrices.
How can I show if $\phi(a)=\phi(b)$ then $a=b$? Same question for onto and operation preserving with matrices.
Thank you!
Best Answer
Hint: For your first question, write down $\phi(a)$ and $\phi(b)$ (go ahead, write down the matrices on a sheet of paper). Now, if those two are equal, what does it tell you?
The other parts are obviously different, but the idea is the same: just look at the matrices involved and use what you know about matrix multiplication (for the last part).