Orbit-stabilizer theorem

For a finite group acting on a set and any point $x$,Indeed, $g{G}_x\mapsto gx$ is a well-defined bijection from the left cosets of the stabilizer to the orbit.

For an algebraic group action, the orbit dimension is the dimension of the group minus that of its stabilizer. For a finite-dimensional representation, the stabilizer is an automorphism group, open in its endomorphism ring, so their dimensions agree. In a quiver representation space, the Ringel form turns the orbit codimension into the dimension of the self-extension group.