Hi,

Let $S=G/K$ be a sphere written as coset space. I know there are just few possibilities for $G$, and $K$ due to the classification of compact connected groups that can be transitive on a sphere.

If $F \subset G$ is a finite fixed point free subgroup, can we write $S/F = G / (F \times K)$??

This should be true for real projective spaces, but for example how about lens spaces $S^{2n-1}/\mathbb Z_q$?

thanks

David