For questions about a function from one group to another that respects the structures of the groups. In symbols, $\varphi$ is a group homomorphisms if for group elements $a$ and $b$, $\varphi(ab)=\varphi(a)\varphi(b)$. Consider also using the broader tags (group-theory) or (abstract-algebra).
A group homomorphism is a function from one group to another that respects the structure of the groups. In symbols, $\varphi\colon G \to H$ is a group homomorphism if for any $a,b \in G$ we have that
$$\varphi(ab) = \varphi(a)\varphi(b)\,.$$
To be clear, the product $ab$ on the left-hand-side is the product in $G$ whereas the product $\varphi(a)\varphi(b)$ on the right-hand-side is in $H$. Here are a few facts about group homomorphisms (hereafter just called homomorphisms) that should reinforce belief in the saying that they respect the structure of the groups involved:
Any homomorphism $\varphi\colon G \to H$ must send the identity of $G$ to the identity of $H$.
Any homomorphisms $\varphi\colon G \to H$ must send inverses to inverses: for any $a \in G$ we have $f(a^{-1}) = f(a)^{-1}\,.$
The composite $\varphi\circ\psi$ of two homomorphisms $\psi\colon K \to G$ and $\varphi\colon G \to H$ is also a homomorphism.
The identity map $\mathbf{1}\colon G \to G$ is a homomorphism.
Noting these last two facts, in the more general language of category theory one would say that group homomorphisms are the morphisms in the category of groups.
