TITLE: Topological groups on the circle $S^1$
QUESTION [5 upvotes]: On the circle $S^1$ there is the usual circle group, i.e. the group isomorphic to $\{e^{i\varphi}\mid \varphi\in[0,2\pi)\}$ with complex multiplication as group operation. This group is a topological group in the sense that $S^1$ is a topological space and the group operation and the inverse are continuous.

Question: Are there other topological groups on $S^1$ essentially different from the circle group, assuming $S^1$ with the standard topology? What about other abelian topological groups?

My question was motivated by this other post. The group described there turned out to be just the usual one.
Two notes:

I am looking for groups involving all of $S^1$, not only a subset. Especially no subgroups of the circle group.
I am looking for groups not isomorphic to the circle group.

REPLY [4 votes]: While Lee Mosher's answer is correct, it is an overkill. Suppose that $G$ is a topological group homeomorphic to $S^1$. Since $G$ is compact, you do not need to quote the solution of Hilbert-V: The Peter-Weyl theorem (which is much simpler) implies that each compact topological group admits a faithful finite dimensional  linear representation and, hence, is a Lie group (Cartan's theorem). Since the Lie algebra ${\mathfrak g}$ of $G$ is 1-dimensional, it is commutative. Hence $\exp: {\mathfrak g}={\mathbb R}\to G$ is a homomorphism (necessarily an epimorphism), hence, $G$ is isomorphic to $U(1)$.