Showing that an action of a group on a set is proper

Question

Let $X$ represent the set = $\{1, 2, 3, 4, 5, 6, 7, 8\}$

Then the elements of the symmetry group (Dihedral Group- $D_8$) permute the elements of the set.

How do I show that the action of $D_8$ on the set is proper?

I understand that the group G must be a topological group (which is continuous) and so it must therefore be infinite.

What I am stuck on is that the group is not infinite.

I have used this website but I am unsure of what I am doing wrong.

Thank you for reading.

Answer 1

I would think that you would include:

G×X->X×X(g,x)|->(gx,x) and relate it to the answer, although I cannot be certain.

Sorry :P