every orbits have same dimension?

Question

For a group G and a set X, let G acts on X. Then we can consider G-orbits. Every G-orbits have same dimensions? And then, dimension of G\X is same as $dim(X) - dim(G-orbits)$?

Answer 1

You are talking of dimension and so the groups are not abstract, some geometric/differential structure must be there. Take $GL(n)$ and look at its action on itself by conjugation. Some orbits are singletons (centralizer element. The scalar matrices). So these have dimension 0. Whereas orbit of a diagonal matrix with distinct entries will have different dimension..

Answer 2

The answer from P Vanchinathan is correct, but you might benefit from a simpler counterexample to the "same dimension" idea. Let $X$ be the Euclidean plane, and let $P$ be a specific point in it (for example, the origin, if your plane comes with coordinates). Let $G$ be the group of rotations of $X$ about $P$ (also known under the fancy name $SO(2)$). Then most of the orbits are the circles with center $P$, and these orbits all have dimension $1$, but there is one orbit consisting of just the single point $P$, and it has dimension $0$. (Given this counterexample, you can probably produce even simpler ones on your own.)