# proving homomorphism versus proving group homomorphism

Quoting " Let $\phi : Z \rightarrow Z$ be given by $\phi(n) = 7n$. Prove that $\phi$ is a group homomorphism. Find the kernel and the image of $\phi$."

Whether proving that we have a homomorphism or proving we have a group homomorphism, isn't it the same proof process?

I prove that it is a group homomorphism by showing that: $$\phi(x \circ y) =\phi(x) . \phi(y)$$ ($x,y \in Z$ and group operation $\circ$ and $.$ are both addition).

Any input is much appreciated.

No, it is not the same. The integers also form a ring, so we could ask for ring homomorphism or group homomorphism - which are two different things. A group homomorphism would require $\phi(x+y)=\phi(x)+\phi(y)$ for the group $(\mathbb{Z},+)$, and a ring homomorphism for the ring $(\mathbb{Z},+,\cdot)$ in addition that $\phi(x\cdot y)=\phi(x)\cdot \phi(y)$ for all $x,y$.