Symmetry Groups
The 2D symmetry groups correspond to the isometry groups, except that symmetry according to O(2) and SO(2) can only be distinguished in the generalized symmetry concept applicable for vector fields.
Also, depending on application, homogeneity up to arbitrarily fine detail in transverse direction may be considered equivalent to full homogeneity in that direction. 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).
These groups also correspond to the one-dimensional symmetry groups, when wrapped around in a circle.
Read more about this topic: Point Groups In Two Dimensions
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