TITLE: Does every homogeneous space allow a group structure?
QUESTION [8 upvotes]: Let $(X,\tau)$ be a homogeneous space, that is for all $x,y \in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$. Is there a group operation $*:X\times X\to X$ such that $(X,*,\tau)$ is a topological group?

REPLY [5 votes]: Another way to get negative answers: a topological group does not have the fixed point property (every map from $X$ to itself has a fixed point), as a translation $x \rightarrow a\ast x$ with $a \neq 1$ does not have a fixed point.
But there are homogeneous compact metric space with the fixed point property, e.g. the Hilbert cube $[0,1]^\mathbb{N}$, see this question for reasons why this is so.