Lecture 14: Symmetry Groups

Lecture 14: Finite and Discrete Groups of Isometries 14 Symmetry Groups So far, in this class, we’ve covered groups and linear algebra. Now, we are looking at groups of symmetries that preserve extra forms of structure. 14.1 Review Last week, we looked at the orthogonal matrices. Defnition 14.1 The orthogonal matrices On are matrices that preserve distance. It is the set T : Rn −→ Rn : |T v| = |v| for all v ∈ Rn . Defnition 14.2 The set Mn of isometries from Rn to itself is {f : Rn −→ Rn : |f (u) − f (v)| = |u − v|}. The orthogonal matrices are the subset of isometries that are linear transformations. In class, we showed that every isometry f is of the form f (x) = Ax + b where A ∈ On and b ∈ Rn . Then, we looked at O2 , the orthogonal matrices in two dimensions. There are two possibilities for a transformation in O2 . • Rotations around 0: these have determinant 1 and are called SO2 . #» • Refections across a line through 0 : these have determinant -1 47 Then the isometries of two-dimensional space, M2 , also ft into several categories.48 • Translations • Rotations around p • Refections across a line • A glide refection49 14.2 Examples of Symmetry Groups Now, we want to add some additional structure to preserve. Guiding Question What isometries of R2 fx some shape inside R2 ? We call the group of such isometries symmetry groups for that shape. Let’s start with a couple examples of shapes and their symmetry groups. 47 The special orthogonal group is quite surprising, since a priori, an isometry could take many diferent forms. 49 A refection in addition to a parallel translation 48 This Lecture 14: Finite and Discrete Groups of Isometries Example 14.3 For a regular pentagon, the group of symmetries are rotations by multiples of lines. This group of symmetries is what we would call discrete.a a This 2π 5 , and refections across will be formalized later on. Next, we look at a group that is not discrete. Example 14.4 For a circle centered at the origin, every rotation or refection will fx it, and so its symmetry group is all of O2 . This group of symmetries is not discrete. We can also look at infnitely large shapes. Example 14.5 For a triangular lattice, certain translations, refections over lines, rotations, and glide refections all preserve it. It is a discrete symmetry group. 14.3 Discrete Subgroups of R From our examples, we see that some symmetry groups are “discrete" and some are not. Guiding Question How can the notion of a discrete group be formalized? We can start with an easier notion, which is a discrete group inside (R, +). Defnition 14.6 A group G ≤ (R, +) is discrete if there exists ε > 0 such that any g ∈ G such that g = ̸ 0 satisfes |g| > ε. Equivalently, for a, b ∈ G and a ̸= b, then it must be true that |a − b| > ε for a discrete group. The discreteness tells us some important information about G. Theorem 14.7 If G ≤ (R, +) is discrete, then G = {0} or G = Zα for some real number α > 0. Lecture 14: Finite and Discrete Groups of Isometries This theorem is very similar to the theorem we had about subgroups of Z, where we showed they were either trivial or of the form kZ. Proof. Assume that G ̸= {0}. Then there is some smallest positive element α ∈ G. To see why it is possible to fnd a smallest element, we start by taking any g > 0 in G. By discreteness, in the interval from [0, g], we have at most g/ε elements of G inside of the interval. We can then pick the smallest one because the set is fnite. We now claim that G = Zα. Why is this true? If 2α < x < 3α for some x ∈ G, then 0 < x − 2α < α, where x − 2α ∈ G, which is a contradiction. 50 14.4 Finite subgroups of O2 So what are all the fnite subgroups of O2 ? Let’s frst try to create some examples to get some intuition about them. Example 14.8 Let x be a rotation by subgroup of O2 . a {1, x, · · · 2π n . Then Cn = ⟨x⟩a , the cyclic group of order n, is generated by x, and is a fnite , xn−1 } Another possible fnite subgroup can be created by expanding Cn a little bit. Example 14.9 #» Let y be a refection across a line ℓ through 0 . Notice that the relations yx = x−1 y, y 2 = e, and xn = e hold, and so any product y a1 xa2 y a3 · · · can be written as xi y j , where 0 ≤ i < n and 0 ≤ j < 2. Then the group generated by x and y is Dn := ⟨x, y⟩ = {e, x, x2 , · · · , xn−1 , y, xy, x2 y, · · · , xn−1 y}, which is called the dihedral group. It has order 2n. ∼ C2 and For n ≥ 3, Dn is the group of symmetries of a regular n−gon.51 The dihedral group for n = 1 is D1 = ∼ C2 ×C2 . For n = 3, D3 = ∼ S3 , and larger dihedral groups can also be studied. for n = 2, D2 = Now, we have two families of fnite subgroups of On , the cyclic groups of rotations, and the dihedral groups. It turns out that these are actually all the fnite subgroups of O2 . This provides yet another classifcation theorem. Let’s start with a simpler version. Theorem 14.10 If a subgroup H ≤ SO2 is fnite, then H is isomorphic to Cn for some n. Proof. Let ρθ be  cos θ sin θ  − sin θ . Then let cos θ S = {θ ∈ R such that ρθ ∈ H}. Under the homomorphism π : θ 7→ ρθ , S = π −1 (H). Since S is a preimage, we know that S is a subgroup of (R, +). If H is fnite, then S must be discrete, and so by Theorem 14.7, S is Zα for some α. Also, 2π ∈ S because a rotation by 2π is the identity in H, and so α = 2π n . So H = Cn . 50 The discreteness guarantees that we can fnd a smallest positive element! This is defnitely not the case for R in general (it is a fundamental property of R that there is no smallest positive element.) 51 In general, if x is a rotation by an angle that is not a rational multiple of 2π, then we do not get a rational group. We would get a non-discrete subgroup of SO2 . Lecture 14: Finite and Discrete Groups of Isometries Theorem 14.11 Any fnite subgroup of O2 is isomorphic to Cn or Dn . Now, we can prove Theorem 15.2. Proof. There are two cases: • Case I. If G ⊆ SO2 , by the above theorem, G ∼ = Cn for some n. • Case II. If G is not a subset of SO2 ,, then take the restriction of the determinant function on O2 to G. It takes det G −−→ {±1}. By the assumption that G isn’t a subset of SO2 , this is surjective. Let det H = ker(G −−→ {±1}). Then, H ⊴ G is a normal subgroup of index 2. So det−1 ({−1}) is a nontrivial coset of H, and so it is Hr for some r ∈ G such that det(r) = −1. Then r must be a refection across some line ℓ.52 Then, it is clear by defnition that H ≤ SO2 , and so H = Cn for some n, and it is generated by some x = 2πρ n , and then we have   2πρ ∼ Dn . G= ,r = n 14.5 More Discrete Subgroups Next, what are the fnite or discrete subgroups of M2 ? Let’s start with a couple of defnitions. Defnition 14.12 A subgroup G ≤ O2 is discrete if there exists some ε > 0 such that all nontrivial rotations in G have angle θ such that |θ| > ε.a a Here, discrete implies fnite, which implies that it is Cn or Dn . Defnition 14.13 A subgroup G ≤ M2 is discrete if there exists some ε > 0 such that all translations in G are by vectors b with |b| > ε, and all rotations in G have angle θ such that |θ| > ε. This ends up being quite a strong constraint on what the discrete subgroups look like, even though there could be lots of diferent possibilities. We’ll talk about this more next time. 52 Note that we have many options for ℓ because any r ∈ Hr generates Hr. In particular, these are all the rotations of ℓ.