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The two-dimensional point groups are important as a basis for the axial three-dimensional point groups, with the addition of reflections in the axial coordinate. They are also important in symmetries of organisms, like starfish and jellyfish, and organism parts, like flowers.


There are two families of discrete two-dimensional point groups, and they are specified with parameter n, which is the order of the group of the rotations in the group.


All of these groups have distinct abstract groups, except for C2 and D1, which share abstract group Z2. All of the cyclic groups are abelian or commutative, but only two of the dihedral groups are: D1 ~ Z2 and D2 ~ Z2×Z2. In fact, D3 is the smallest nonabelian group.


where SO(2) has any possible θ. Not surprisingly, SO(2) and its subgroups are all abelian; addition of rotation angles commutes.





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