I was reading an introduction to something called Mackey Theory in a Representation Theory pdf and I came across the following statement: Does this mean that the double coset [itex]H\backslash G / K[/itex] can be understood to be a set of orbits? That is, a set of orbits of the left action of the product group [itex]H \times K[/itex] on [itex]G[/itex]? What I dont get is this: I've also read that the double coset is the set of orbits for the left action of [itex]H[/itex] on the coset [itex]G/K[/itex] induced by the action [itex](h,g)\mapsto hg[/itex] of [itex]H[/itex] on [itex]G[/itex]. So which definition of the double coset is correct? Or are they the same?