I would guess the complex numbers under addition. You are quotienting out a copy of the reals (under +) to get a copy of the reals (under *). Apparently the group you start with is one you are `familiar' with. Well, you only know a couple of groups which are uncountable...

I hope that helps for the moment!

EDIT: Although so you will have to look quite hard for this copy of the reals, it certainly isn't obvious...

EDIT2: Powers seems to work. You are wanting to turn addition into multiplication, so powers seem to be a sensible choice.

a, b) \mapsto e^{a+b}" alt="\phia, b) \mapsto e^{a+b}" />. Clearly and .

You now just need to prove that the kernel is isomorphic to the reals. However, the kernel is the set which is isomorphic to the reals under the isomorphism .