The answer is no in general, but this is a rather deep fact.

Theorem: (Nikolov, Segal) If $G$ is any compact Hausdorff topological group, then every finitely gen- erated (abstract) quotient of $G$ is finite.

N. Nikolov and D. Segal, Generators and commutators in finite groups; abstract quotients of compact groups, arXiv, http://arxiv.org/abs/1102.3037

The answer is no in general, but this is a rather deep fact.

Theorem: (Nikolov, Segal) If $G$ is any compact Hausdorff topological group, then every finitely generated (abstract) quotient of $G$ is finite.

N. Nikolov and D. Segal, Generators and commutators in finite groups; abstract quotients of compact groups, arXiv, http://arxiv.org/abs/1102.3037