Homomorphisms: What am I missing here?

Question

I am considering a homomorphism $\phi:\Bbb{Z}_{12} \to \Bbb{Z}_{30}$ by defining $\phi(1)=7 \pmod{30}$ - mapping a generator of $\Bbb{Z}_{12}$ to a generator of $\Bbb{Z}_{30}$. Then noting that $1 \equiv 25 \equiv5 \cdot 5 \pmod{12}$, but then $\phi(5\cdot 5)=\phi(5)+\cdots +\phi(5)=5\phi(5)=5\cdot 5\phi(1)=5\cdot 5\cdot 7=175=25 \pmod{30}$ imply $\phi(1)=\phi(5\cdot 5)=25 \pmod{30}$. Which cannot be since $\phi$ must be a function. What am I missing?

Answer 1

Your calculation is simply a proof that there does not exist any homomorphism $\phi:\Bbb{Z}_{12} \to \Bbb{Z}_{30}$ such that $\phi(1)=7 \pmod{30}$.

Note that you never actually defined the map $\phi$ or proved that your definition gives a homomorphism. You just defined what you wanted $\phi(1)$ to be, and assumed that $\phi$ was a homomorphism in order to compute other values of $\phi$.