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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?

Peter_Pan
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    Do you really talk about isomorphisms between two isometries? – pancini Sep 12 '19 at 07:21
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    So you are talking about isomorphism classes of groups. An isomorphism class is the collection of all groups which are isomorphic (to each other, or to a particular representative if you want). In the example you gave, the isometries of the triangle and the isometries of the square form two groups which are not isomorphic, so they are not in the same isomorphism class. – pancini Sep 12 '19 at 07:39
  • To add to @ElliotG's comment, one reason that $D_3$ and $D_4$ can't be isomorphic is that $D_3$ has an element of order $3$ and $D_4$ has no elements of order $3$. – Charles Hudgins Sep 12 '19 at 07:49
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    Do you mean to put isometries into isomorphism classes, or groups of isometries into isomorphism classes? – Joppy Sep 12 '19 at 10:29

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"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)

Dietrich Burde
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