Proving a group homomorphism between two groups with different operations

Question

Here is a rather simple algebra question: Suppose $(G,\star)$ and $(H,\circ)$ are groups and $\varphi$ is a map from $G\rightarrow H$. Am I correct in understanding that to show $\varphi$ is a homomorphism is to show that for any $g,h\in G$, $$\varphi(g\star h) = \varphi(g)\circ\varphi(h),$$ e.g. the map factors with respect to the operation on $H$? Since most textbooks notate the operations in both groups by placing the symbols next to each other this is often written as $$\varphi(gh) = \varphi(g)\varphi(h),\ g,h\in G,$$ which is a bit confusing (for someone without much talent in mathematics, like me) until it is dissected...

Answer 1

Yes, that is what $\varphi(gh)=\varphi(g)\varphi(h)$ means. If writing the concrete binary operation makes it easier for you to understand what's going on, then there is no harm in writing it. Just be aware that many do not.