In my professor's lecture notes, I've noticed that sometimes the $\circ$ symbol is used when operations are combined, but at other times multiplication is referred to , for example in the definition of a homomorphic relationship:

$$\phi:G\rightarrow H,\,\ \ \ g,h\in G, \ \ \ \phi(g), \phi(h)\in H$$
$$\phi(g\star h) = \phi(g)\cdot \phi(h)$$

Is there any difference between the two? To me it seems like the multiplication of $g$ and $h$ (e.g. in a symmetry group) simply represents two successive operations applied to an object, which is essentially the definition of function composition.

EDIT: I was perhaps a bit unclear here; my question was not directed at the difference between $\star$ and $\cdot$, it was the difference between composition and the group operation. I'll leave it as it is now, so that BobaFret and Foobaz John's answers make sense.

[Here](https://math.stackexchange.com/questions/1229008/why-do-we-think-of-group-compositions-as-multiplication)'s a similar question, but that question deals with why we refer to the operation as multiplication, not whether it is analogous to composition (as far as I can tell).