Consider the group $D_4$. Give examples of $D_4$ acting on a set.
Attempt: So $|D_4| = 8$. I have come up with a few, but I was wondering what some people here thought. First one we came up with in class was $D_4$ acting on the set of vertices of a square. Am I correct in saying $D_4$ acts on this set because there are eight symmetries of the square and 8 elements in $D_4$. So each element corresponds to a symmetry.
I think I can extend this to an octagon, which has 8 faces, and so each element can correspond to a face.
Another one I came up with was the set of edges on a cube. Each element in $D_4$ can correspond to an edge.
Is my reasoning correct above for why $D_4$ could act on these sets? Can anybody suggest others?