TITLE: Is there a compact group of countably infinite cardinality?
QUESTION [27 upvotes]: Apologies for the very simple question, but I can't seem to find a reference one way or the other, and it's been bugging me for a while now.
Is there a compact (Hausdorff, or even T1) (topological) group which is infinite, but has countable cardinality? The "obvious" choices don't work; for instance, $\mathbb{Q}/\mathbb{Z}$ (with the obvious induced topology) is non-compact, and I get the impression that profinite groups are all uncountable (although I might be wrong there). So does someone have an example, or a reference in the case that there are no such groups?

REPLY [2 votes]: It is a well known fact (see for example Hodel´s article on the Handbook of Set Theoretic Topology) that the cardinality of an infinite compact homogeneous space is always a power of $2$. Topological groups are homogeneous (as it is pointed out in Yemon´s answer) and therefore any infinite compact group has the size of the continuum or bigger.