Sets of functions which convolve “around” in a group?

I have read some baby's first basic abstract algebra with some group theory. As far as I understand if we have a

1. Set of any kind of mathematical object and then
2. Some binary operator which makes sense for these objects and which outputs new objects.
3. Follows the group axioms

Then we have a group!

So to my question, if we imagine as a set we pick a bunch of functions and as binary operator we pick convolution... Do there exist any famous such sets which with together it can create a group?