Relationship between orbits and cosets

potmobius

How are orbits and cosets related? Are all orbits cosets? Are all cosets orbits? Also, what exactly are G-sets and G-equivariant sets?

lavinia

A group permutes cosets by group multiplication.

An orbit of a point in a group action is just all of the points that the point reaches under the action.

But groups can act on may things and they certainly do not have to be cosets.

However, suppose that a group acts transitively on a set. (This means that there is only one orbit in the group action.)

Then the stabilizer of any point is a subgroup of the group (can you prove that?) and the coset space is isomorphic to the original set as a G-space. this you should prove for yourself.

So the group action can be represented as the action on the set of cosets of the stabilizer subgroup.