The symmetry group of a square belongs to the family of dihedral groups, Dn (abstract group type Dihn), including as many reflections as rotations. The infinite rotational symmetry of the circle implies reflection symmetry as well, but formally the circle group S1 is distinct from Dih(S1) because the latter explicitly includes the reflections. An infinite group need not be continuous; for example, we have a group of all integer multiples of rotation by 360°/, which does not include rotation by 180°. 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. Cn and Dn for n = 1, 2, 3, 4, and 6 can be combined with translational symmetry, sometimes in more than one way. Thus these 10 groups give rise to 17 wallpaper groups. 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. Combinations with translational symmetry E(2) is a semidirect product of O(2) and the translation group T. In other words, O(2) is a subgroup of E(2) isomorphic to the quotient group of E(2) by T: O(2) E(2) / T There is a "natural" surjective group homomorphism p : E(2) → E(2)/ T, sending each element g of E(2) to the coset of T to which g belongs, that is: p (g) = gT, sometimes called the canonical projection of E(2) onto E(2) / T or O(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