Group theory - composition vs. multiplication 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'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).

The author seems to be stressing the fact that $\star$ and $\cdot$ are two different group operations. $\star$ is an operation on $G$ while $\cdot$ is an operation on $H$. Usually though we just write
$$
\phi(gh)=\phi(g)\phi(h)
$$
where the operation on the left is in $G$ and operation on the right is in $H$. 
