As I understand things so far, the ECDSA scheme used by Bitcoin uses a group G generated by a point on the elliptic curve y^2 = x^2 + 7.

Now G is a cyclic group of finite order n and is, hence, isomorphic to the additive group Z/nZ.

My question is: why not work within Z/nZ directly then? I am assuming there are certain security benefits of using G but I am lacking the general overview to understand why.