Symmetry group

A frequent notation for the symmetry group of an object X is G = Sym(X).

This article mainly considers symmetry groups in Euclidean geometry, but the concept may also be studied for more general types of geometric structure.

(A pattern may be specified formally as a scalar field, a function of position with values in a set of colors or substances; as a vector field; or as a more general function on the object.)

The above is sometimes called the full symmetry group of X to emphasize that it includes orientation-reversing isometries (reflections, glide reflections and improper rotations), as long as those isometries map this particular X to itself.

Discrete symmetry groups come in three types: (1) finite point groups, which include only rotations, reflections, inversions and rotoinversions – i.e., the finite subgroups of O(n); (2) infinite lattice groups, which include only translations; and (3) infinite space groups containing elements of both previous types, and perhaps also extra transformations like screw displacements and glide reflections.

However, this excludes for example the 1D group of translations by a rational number; such a non-closed figure cannot be drawn with reasonable accuracy due to its arbitrarily fine detail.

D2, which is isomorphic to the Klein four-group, is the symmetry group of a non-equilateral rectangle.

Within each of these symmetry types, there are two degrees of freedom for the center of rotation, and in the case of the dihedral groups, one more for the positions of the mirrors.

In crystallography, only those point groups are considered which preserve some crystal lattice (so their rotations may only have order 1, 2, 3, 4, or 6).

The continuous symmetry groups without a fixed point include those with a screw axis, such as an infinite helix.

Conversely, specifying the symmetry group can define the structure, or at least clarify the meaning of geometric congruence or invariance; this is one way of looking at the Erlangen programme.

Add some patterns such as arrows or colors to X so as to break all symmetry, obtaining a figure X# with Sym(X#) = {1}, the trivial subgroup; that is, gX# ≠ X# for all non-trivial g ∈ G. Now we get: Normal subgroups may also be characterized in this framework.

The symmetry group of the translation gX + is the conjugate subgroup gHg−1.

Thus H is normal whenever: that is, whenever the decoration of X+ may be drawn in any orientation, with respect to any side or feature of X, and still yield the same symmetry group gHg−1 = H. As an example, consider the dihedral group G = D3 = Sym(X), where X is an equilateral triangle.

We may decorate this with an arrow on one edge, obtaining an asymmetric figure X#.

Letting τ ∈ G be the reflection of the arrowed edge, the composite figure X+ = X# ∪ τX# has a bidirectional arrow on that edge, and its symmetry group is H = {1, τ}.

This subgroup is not normal, since gX+ may have the bi-arrow on a different edge, giving a different reflection symmetry group.

Then H is normal, since drawing such a cycle with either orientation yields the same symmetry group H.