Klein 4-group is a symmetry group of rectangle (or rombus). And as far as I understand it is not isomorphic to Dihedral group of order 4. Because Dihedral group of order 4 is a group of rotations of square.

If this is true then why Wiki says that they are isomorphic:

The Klein four-group is the smallest non-cyclic group. It is however an >abelian group, and isomorphic to the dihedral group of order (cardinality) 4

What am I missing here?


You are confusing 'the dihedral group with 4 elements' with 'the dihedral group that is the symmetry group of the square'. The dihedral group with 4 elements is the set of symmetries of a bigon (i.e. a 'polygon' with 2 vertices, which can be thought of as two curved lines joined together) or of a rectangle (thanks to Akiva Weinberger).

The dihedral group that is the symmetry group of a square is of order 8 and usually is denoted $D_4$.

  • 1
    $\begingroup$ It's best to imagine the 2-gon as having two curved sides, here, so that it has an "area". Thus, the symmetries are the identity, rotating, flipping horizontally, and flipping vertically. (There are only two symmetries of an actual line segment, I believe — identity and rotating.) $\endgroup$ – Akiva Weinberger Nov 5 '15 at 3:57
  • $\begingroup$ Thank you, guys, I got it now! $\endgroup$ – IgorStack Nov 5 '15 at 4:01

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