Homogeneous space

Summary

A homogeneous space for a group $G$ is a space on which $G$ acts transitively, so that all points look alike. In algebraic geometry, it is a variety with a transitive action of an algebraic group. Such a variety is a quotient $G / H$ by the stabilizer $H$ of a point, at least in characteristic $0$. Projective spaces, Grassmannians and flag varieties are homogeneous spaces.