Examples of the dihedral group $D_4$ acting on sets

Question

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?

Many thanks

Answer 1

To define a group action, you need:

1) A group $G$ (In you case, $G = D_4$ is fixed.)

2) A set $X$ (you gave several suggestions, like the set of vertices of a square).

3) A multiplication rule $G\times X \to X$ satisfying the axioms of a group action.

In your suggestions, I'm missing 3).

Answer 2

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.

No, this is not correct: there are other eight-element groups, e.g. the cyclic group of order 8 or the quaternion group, that don't correspond to symmetries of the square. It is crucial that there is a pairing of group elements with symmetries in such a way that they compose meaningfully, i.e. that the symmetry paired with $xy$ is the symmetry paired with $y$ followed by the symmetry paired with $x$.

To find more group actions, recall that a group action is faithful when the only element that doesn't do anything is the identity, and in particular group actions do not need to be faithful – not all of the elements of the group need to act in an interesting way.

Answer 3

The group $D_4$ acts on $ℝ^2$. As the members are bijections from $$ℝ^2→ℝ^2 : x↦ρ^iσ^j :i ∈\{0,1,2,3 \},j∈\{0,1\} $$ where $ρ$ is the standard rotation and $σ$ a relection.

Define the group action as the map $$D_8 ×ℝ^2→ ℝ^2:(ρ^iσ^j,x) ↦ρ^iσ^j.x=ρ^iσ^j(x)$$

Let $x ∈ℝ^2$ , then $e.x=e(x)=ρ^0σ^0(x)=x$.
And let $g_1,g_2∈D_4$ then,
$$g_1.(g_2.x)=g_1.(g_2(x))=g_1( g_2 (x))=g_1 \circ g_2 (x)$$