Point groups in two dimensions

In geometry, a two-dimensional point group or rosette group is a group of geometric symmetries (isometries) that keep at least one point fixed in a plane. Every such group is a subgroup of the orthogonal group O(2), including O(2) itself. Its elements are rotations and reflections, and every such group containing only rotations is a subgroup of the special orthogonal group SO(2), including SO(2) itself. That group is isomorphic to R/Z and the first unitary group, U(1), a group also known as the circle group.

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.

Intl refers to Hermann-Mauguin notation or international notation, often used in crystallography. In the infinite limit, these groups become the one-dimensional line groups.

If a group is a symmetry of a two-dimensional lattice or grid, then the crystallographic restriction theorem restricts the value of n to 1, 2, 3, 4, and 6 for both families. There are thus 10 two-dimensional crystallographic point groups:

The groups may be constructed as follows:

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.

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