How to classify discrete groups of motions

Question

I'm studying chapter 5 of Artin's Algebra, and it talks about the classification of the discrete groups of motions(in the plane), and that there are only 17 types. I thought about what it means for two discrete groups to be equal, and it didn't seem like group isomorphism was the right criteria to classify these groups. For example, I can't think of a way to prove that all of these groups are non-isomorphic. So what is the correct criteria? It is just group isomorphism? If it is not, why wouldn't Artin include that in his book?

Answer 1

Yes, the classification is up to isomorphism of abstract groups. All $17$ groups can be written down by a presentation and an explicit semidirect product. This makes it easy to see that the groups are non-isomorphic. For example, the first group is $\Bbb Z^2$, where we have only translations, and the last group is $D_6\ltimes \Bbb Z^2$, where we have the point group $D_6$, the dihedral group.

References:

What are the algebraic structures of the wallpaper groups?

Proof that there exist only 17 Wallpaper Groups (Tilings of the plane)