Lecture 17: Group Actions

Lecture 17: Group Actions 17 Group Actions Today, we will discuss group operations or actions68 on a set. Guiding Question How can a group be seen as a group of transformations? 17.1 Review Last time, we fnished talking about (discrete) subgroups of isometries of the plane. Finite subgroups of M2 are isomorphic to Cn or Dn , and there are only fnitely many isomorphism classes of infnite discrete subgroups of M2 .69 It is also possible to go up a dimension and classify discrete subgroups of isometries of R3 , although it is more complicated; there are 200 or so. 17.2 Motivating Examples The idea of a group action will generalize and make more abstract an idea that has been present throughout the class so far. Let’s start with the following motivating example. Example 17.1 (GLn ) Given g ∈ GLn (R) and a column vector v ∈ Rn , the matrix g can be seen as a transformation on Rn , taking v 7→ g(v) ∈ Rn . The data of GLn (R) acting on Rn can be packaged together by a map GLn (R)×Rn −→ Rn (g, #» v ) 7−→ g( #v»). The same principle applies to Sn , the group of permutations on {1, · · · , n}. Example 17.2 (Sn ) The symmetric group Sn can also be viewed as acting on a set. More or less by defnition, given a number between 1 and n, and a permutation, it’s possible to spit out the result of permutation acting on that number. So Sn permutes the set [n] = {1, · · · , n}. This gives us another mapping encoding this information: Sn ×{1, · · · , n} −→ {1, · · · , n} (σ, i) 7−→ σ(i). Our last example is one we have been considering for the past few lectures. Example 17.3 The set M2 , isometries of 2-space, acts on R2 : given some vector in the plane and some isometry, the isometry will return some other vector in the plane. This information is again encoded by a mapping M2 (R)×R2 −→ R2 (f, ⃗x) 7−→ f (⃗x). 68 They are diferent terms for the same idea. Artin uses group operations, while Professor Maulik prefers to call them group actions. 69 In fact, with any metric space, which is a set with some distance on it (as discussed in 18.100, for example), it’s possible to consider isometries, distance-preserving transformations, in the same way as we considered the plane R2 . Depending on the metric space, the groups can look very diferent! One example of this is the hyperbolic plane, which is the upper half-plane of R2 with a non-Euclidean metric, or distance, on it, and the discrete subgroups of isometries on it. There are infnitely many discrete subgroups of isometries on it, even though it is 2-dimensional, just like R2 . The question of why it is so diferent from the R2 case is really a geometry question, rather than an algebra question. Lecture 17: Group Actions 17.3 What is a group action? These are all examples of group operations on a set, and they motivate the following defnition of a group operation in general. Defnition 17.4 Given a group G and a set S, a group actiona on a set S is a mapping G×S −→ S (g, s) 7−→ gs. It must satisfy the following axioms: • The identity maps every element of the set back to itself: es = s for all s ∈ S.b • The mapping must respect the group multiplication: (gh)s = g(hs) for s ∈ S and g, h ∈ G.c a or group operation corresponds to the identity multiplication rule. c This corresponds to the associativity rule. b This Essentially, given an element g ∈ G and s ∈ S, the mapping returns another element of S depending on g, and the mapping must respect the group structure on G. All of the groups that we have seen so far show up as symmetries of some set, maybe preserving some extra structure, so all the groups that we usually think about already come with an action on some set S. Furthermore, a group G can act on many diferent sets at the same time in diferent ways, which gives insight into the group itself. Let’s look at a couple of examples. Example 17.5 (S4 ) The symmetric group S4 , permutations on 4 elements, acts on S = {1, 2, 3, 4}. It can also act on a diferent set, T = {unordered pairs in S} = {(12), (13), (14), (23), (24), (34)}. The set T has 6 elements, and S4 acts on T as well as acting on S. Given a permutation σ ∈ S4 , and an unordered pair {i, j}, it acts by taking σ({i, j}) = {σ(i), σ(j)} for a permutation σ ∈ S4 . So the group action on S leads to a diferent group action on a diferent set, T. The existence of a group action on a given set actually yields a lot of information about the group G, as will be explored in the next few lectures. Let’s see a diferent example. Example 17.6 (D2 ) Let G = D2 , which contains rotation by π as well as refection across the x-axis (and then refection across the y-axis.) As a subgroup of O2 ,a , D2 will act on all of R2 . It also acts on the set S consisting of the vertices of a square and a diamond, as well as the center. a 2×2 orthogonal matrices A group G can also act on itself viewed as a set. Lecture 17: Group Actions Example 17.7 Given a group G, there is a mapping G×G −→ G (g, g ′ ) 7−→ gg ′ , and this is a valid group action. When G acts on itself, the frst G in G×G is seen as a group, while the second G is seen as a set, since the axioms of a group action don’t care about the group operation on the second instance of G. Let’s see one last example. Example 17.8 Taking a vector space V over a feld F, the group F × , the nonzero elements of the feld, which is a group with respect to multiplication, acts on V by scaling: F × ×V −→ V (a, ⃗v ) 7−→ a⃗v . Scaling by nonzero scalars defnes a group operation! It satisfes each of the axioms. Student Question. What type of element is g(s)? Answer. It depends on what S is! It is the type of element that is in S. Two group actions of G on S and S ′ might not have anything to do with each other, other than the fact that they both involve G; G can act on wildly diferent types of sets, and show up in diferent contexts. Say we fx an element g ∈ G, we can defne the group action of g on S, a mapping τg : S −→ S sending s 7−→ g(s).70 We can show that τg is a bijection from S to itself because it has an inverse map, τg−1 , coming from the fact that g is invertible. Because τg is a bijection, it actually permutes the elements of S, and so it is a permutation of S. Thus, each element of G can be mapped to a permutation by a map τ : G −→ Perm(S), which takes g 7−→ τg ∈ Perm(S). From the group action axioms, τ is a group homomorphism. In Example 17.6, D2 is acting on a set with |S| = 9, so there exists a homomorphism from D2 −→ Perm(S) = S9 . Note that τ does not have to be injective; there may be some action g ∈ G such that g = ̸ e but G fxes each s ∈ S, which would make τ (g) the identity permutation. 17.4 The Counting Formula Defnition 17.9 Given s ∈ S, the orbit of s is Os = Gs := {gs : g ∈ G} ⊆ S. For instance, in Example 17.6, there are several orbits of diferent sizes. The top and bottom vertices of the diamond are in the same orbit (size 2), the left and right vertices of the diamond are in the same orbit (size 2), all the vertices of the square are in the same orbit (size 4), and the origin is in an orbit by itself (size 1), just by applying each of the group elements to an element of the set. Defnition 17.10 The group G acts transitively on S if S = Os for some s ∈ S. For example, Sn acts transitively on {1, · · · , n}, since given an element i ∈ {1, · · · , n}, there is some permutation mapping it to any other element i′ . 70 This notation is not standard and may not correspond with the textbook. Lecture 17: Group Actions Student Question. Does this have to be true for all s ∈ S, or just one? Answer. If it is true for one s ∈ S, it is true for all s ∈ S. Try checking it! So a transitive group action is one where there is only one orbit consisting of the entire set S; in particular, any element of s can be carried to any other element when acted on by some g ∈ G. Defnition 17.11 The stabilizer of s is Gs = StabG (s) := {g ∈ G : gs = s}, and it is a subgroup of G. For Example 17.6, the top and bottom vertices of the diamond are stabilized by the refection across the y-axis, whereas the stabilizer group of a vertex of the square is just the identity element. Proposition 17.12 The orbits of G form a partition of S.a In particular, S is the disjoint union of the orbits: S = ⨿Oi where Oi ∩ Oj = ∅. a The set can be cut into non-overlapping pieces by the orbits. Proof. The orbits clearly cover S, since every element s ∈ S is also an element of Os , its own orbit. Also, they are disjoint. If Os ∩ Os′ ̸= ∅, then there is some element in their intersection t = gs = g ′ s′ . Then s = (g −1 g ′ )s′ , which is in Os′ . So every element of Os is in Os′ , and by the same logic Os′ ⊆ Os . Then Os = Os′ . So if two orbits have nonempty intersection, they are in fact the same orbit. For a fnite set, the size of S can be obtained from the sizes of the orbits. Corollary 17.13 If S is a fnite set, and O1 , · · · , Ok are the orbits, then |S| = k X i=1 since each of the orbits cover S exactly. In Example 17.6, this gives 9 = 4 + 2 + 2 + 1. Guiding Question What does each orbit look like? For this, we use the notion of a stabilizer of an element. |Oi |, Lecture 17: Group Actions Proposition 17.14 Fix some s ∈ S and let H := Stab(s). Then there exists a bijection ε from the quotient group G/H to the orbit of s, Os . It takes ε G/H − → Os gH 7→ gS. Proof. Consider g and γ in G. Then their cosets map to the same element if gs = γs, which is equivalent to saying that g −1 γs = s. Since H is the stabilizer of S, this means that g −1 γ ∈ H; equivalently, γ ∈ gH. Since each of these conditions were equivalent conditions, gs = γs if and only if γ ∈ gH, and thus ε must be bijective: two elements in G/H map to the same element in Os if and only if they are the same element. Corollary 17.15 (Counting Formula for Orbits) As a result, the number of cosets of H, which is the order |G/H|, is equal to the size of the orbit of s, since there is a bijective correspondence between them. So |Os | = [G : Stab(s)]. In particular, the size of the orbit of any element |Os | divides |G| when G is a fnite group. We have |Os | · |Stab(s)| = |G|. These theorems are similar to the Counting Formula and Lagrange’s Theorem from Chapter 2. In particular, let C be the set of left cosets of a given subgroup H. Then G acts on C; an element g ∈ G takes C 7−→ gC. Every coset can be mapped to any other coset by some element of G. For example, g1 H is mapped to g2 H by g2 g1−1 ∈ G. So there is only one orbit, the entire set C. The stabilizer of the identity coset, which is eH = H, is Stab(eH) = H, because some element g ∈ G carries h ∈ H to h′ ∈ H if and only if gh = h′ , which implies that g = h′ h−1 ∈ H. Thus, the Orbit-Stabilizer Theorem states that |G| = |H|[G : H], since |H| is the stabilizer of the identity in G/H and [G : H] is the size of the identity orbit. Lecture 17: Group Actions Example 17.16 Consider the subgroup G ≤ SO3 consisting of rotational symmetries of a cube centered at the origin. • Let S be the set of faces of the cube; it has order 6 since there are 6 faces. For every face of the cube, there is some element in G mapping it to any other face in the cube (G acts transitively on the faces), so the orbit of a given face is the set of all the other faces, which is S. The stabilizer StabG (face) = C4 , since a given face, which is a square, is preserved by rotation by π/2 around the axis through the center of the face. Then |G| = |S| · |StabG (face)| = 6 · 4 = 24. • Similarly, any vertex can be mapped to any other vertex by some element of G. The stabilizer StabG (vertex) = C3 , since a vertex is preserved under rotation by 2π/3 around the axis from the vertex to the opposite vertex. Again, |G| = |{vertices}| · |StabG (vertex)| = 8 · 3 = 24. • Again, G acts transitively on the set of edges. The stabilizer of an edge is StabG (edge) = C2 . Then |G| = |{edges}| · |StabG (edge)| = 12 · 2 = 24.