In the formulas:
$ \begin{align*} S^{n-1} &\cong \frac{O(n)}{O(n-1)} \\ G(k,n) &\cong \frac{O(n)}{O(k) \times O(n-k)} \end{align*} $
what do the notations $\frac{O(n)}{O(n-1)}$ and $\frac{O(n)}{O(k) \times O(n-k)}$ mean? I know the definition of quotient topology, but in that case the denominator must be either a surjective function or an equivalence relation like $X / \sim$. But what does this notation mean when both numerator and denominator are spaces? (In the part of my book where these formulae were mentioned, they used terms like 'action of a group', 'transitive actions' - I read these terms on Wikipedia, but I can't see the relation between these terms and the above formulae.)