The definition of a one-parameter subgroup of a topological group $G$ is given as a particular group homomorphism $\phi: \Bbb{R} \rightarrow G$. I'm not sure I understand the terminology. Why do we say that the homomorphism is the subgroup rather than the image being the subgroup?
If I understand correctly, it has something to do with the parameterization being extra information attached to the group. For example, $t \mapsto e^{it}$ and $t \mapsto e^{i\omega t}$ (for $\omega \in \Bbb{R}_{\ne 0}$) have the same image, but if we imagine that the elements of $\Bbb{C}^\times$ "remember where they came from," then we can distinguish the resulting groups. Presumably this has some use that I will learn soon.
I don't think I've come across this kind of pattern before, where we can attach additional information to an algebraic structure without thinking of it as a new kind of structure. Is this common?
Intuitively you can think of the parameter $t$ as a time parameter. That means the homomorphism tells you how "fast" the one-parameter subgroup is going, and its image doesn't. I like to think specifically about one-parameter subgroups of $SO(3)$, which correspond to "motions" of the form "rotate in this direction this quickly."
Remembering the homomorphism also lets you differentiate it to get an element of the Lie algebra. It just turns out to be a really good idea. The image is not quite enough information.