You can see a group and similar structures as sets of rules an object needs to follow to be considered a group or whatever. Conceptually, a group is anything that behaves like a group. It could be a dog! So, the operator can be anything you want as long as the indicated properties hold. It's like a generic API that lets you use whatever concrete type you want as long as it conforms to certain rules.

edit: What I mean is that, as a consequence, the symbol used is not really important.