An isomorphism that takes Z12 (integers modulo 12 under addition) to Z13* (integers modulo 13 under multiplication) I'm having a hard time finding an isomorphism that takes the integers in $\mathbb{Z}_{12}$ (those integers modulo 12 under addition) to the integers in $\mathbb{Z}_{13}^{*}$ (those integers modulo 13 that are relatively prime to 13 under multiplication)
The hint says to start with $\phi(1)$, where $\phi$ is the isomorphism, and try various $\phi(1)$ until that works.
Attempt:
I've started off by trying to work out what exactly the problem is asking.
If I have $7+9$ in $\mathbb{Z}_{12}$ that's equal to 4. So what does that mean for $\mathbb{Z}_{13}$? $7 \cdot 9=11$ in $\mathbb{Z}_{13}^{*}$. So do I want something that takes 4 to 11 if that was the case?
 A: The two groups are both cyclic, which means that if there is an isomorphism it has to be a homomorphism that sends one generator to another. 
Finding a generator in the former group is pretty simple: $1$ will do the trick. 
The question now is: What is a generator in the latter group?
Can you take the problem from here?

Now that the problem has been solved, it might be interesting to note how you can move between these two groups using our isomorphism to solve some problems from number theory.
For example: What is $2^{25}$ mod $13$?
Answer: Using our isomorphism, we can send this back to $(\mathbb{Z}_{12}, +)$, where it is $25$ mod $12$, which is the same as $1$ mod $12$. We can now send this back where it came from, where it is $2^1$ mod $13$.
And there you have it: $2^{25}$ mod $13$ $\equiv$ $2$ mod $13$.
There are other (even simpler) ways to tackle this example, but the approach above might give you an idea about how Group Theory and Number Theory intersect; an early and important result one comes across in a first course on Abstract Algebra is often Fermat's Little Theorem, which is quite easily proved using ideas from Group Theory (namely, Lagrange's Theorem).
