I don't know anything about representation theory, but the way you describe it I thought "how's this different from a homomorphism." All elements of a group (e.

I don't know anything about representation theory, but the way you describe it I thought "how's this different from a homomorphism." All elements of a group (e.g. any x, y in your example), are invertible by the definition of a group. The wikipedia sentence definition makes more sense to me: "a "representation" means a homomorphism from the group to the automorphism group of an object" This seems natural, since if you take _any_ algebraic structure (e.g. a ring), then its automorphism group is a group. Which is a reason why group theory is worth studying - since _any_ algebraic structure can be studied using group theory by considering its automorphism group. (Also, an automorphism group is just the set of symmetries, so group theory can be described as the study of symmetry).