Basic question, but I don't understand what an isomorphism class is, in particular I have a homework question about isomorphism classes of groups that can arise as isom($\mathbb{R}^2$).
For example, I think that if I'm talking about isomorphism classes of groups of isometries in $\mathbb{R}^2$, an isomorphism class consists of all isometries that are the same on different sets.
So, if I have equilateral triangle and a regular square, the rotations an isomorphism class. But there are different numbers of rotations in $D_3$ and $D_4$, how can there be an isomophism between these two groups if they have different numbers of elements?
"The homework is about the isomorphism classes of groups that can arise as $\operatorname{Isom}(\Bbb R^2)$". The answer is that $$ \operatorname{Isom}(\Bbb R^2)\cong O_2(\Bbb R)\ltimes \Bbb R^2, $$ so that there is only one isomorphism class.
On the other hand, there are the famous $17$ isomorphism classes of wallpaper groups, being discrete and cocompact subgroups of $\operatorname{Isom}(\Bbb R^2)$.
References:
What is the isometry and isometry group?
Proof that there exist only 17 Wallpaper Groups (Tilings of the plane)