TITLE: How many group structures make $S^1$ a topological group?
QUESTION [6 upvotes]: Let $S^1$ be the subspace of $R^2$ given the usual topology. How many group structures make $S^1$ a topological group?

REPLY [2 votes]: If $S^1$ is a topological group, it is, in a unique way, a Lie group in a compatible way, and with that structure, isomorphic to the usual $S^1$ by the structure theory of compact abelian Lie groups.