Point groups in two dimensions

The continuous cyclic group SO(2) or C∞ and its subgroups have elements that are rotation matrices: where SO(2) has any possible θ.

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

For discrete cyclic groups Cn, elements Cnk = R(2πk/n) The continuous dihedral group O(2) or D∞ and its subgroups with reflections have elements that include not only rotation matrices, but also reflection matrices: where O(2) has any possible θ.

These groups fall into two distinct families, according to whether they consist of rotations only, or include reflections.

Depending on its application, homogeneity up to an arbitrarily fine level of detail in a transverse direction may be considered equivalent to full homogeneity in that direction, in which case these symmetry groups can be ignored.

This greatly simplifies the categorization: we can restrict ourselves to the closed topological subgroups of O(2): the finite ones and O(2) (circular symmetry), and for vector fields SO(2).

Its kernel is T. For every subgroup of E(2) we can consider its image under p: a point group consisting of the cosets to which the elements of the subgroup belong, in other words, the point group obtained by ignoring translational parts of isometries.

For every discrete subgroup of E(2), due to the crystallographic restriction theorem, this point group is either Cn or of type Dn for n = 1, 2, 3, 4, or 6.

For a given hexagonal translation lattice there are two different groups D3, giving rise to P31m and p3m1.

If the isometry group contains an n-fold rotation then the lattice has n-fold symmetry for even n and 2n-fold for odd n. If, in the case of a discrete isometry group containing a translation, we apply this for a translation of minimum length, then, considering the vector difference of translations in two adjacent directions, it follows that n ≤ 6, and for odd n that 2n ≤ 6, hence n = 1, 2, 3, 4, or 6 (the crystallographic restriction theorem).