Groups acting on sets 1 Definition and examples Most of the groups we encounter are related to some other structure. For example, groups arising in geometry or physics are often symmetry groups of a geometric object (such as Dn) or transformation groups of space (such as SO3). The idea underlying this relationship is that of a group action: Definition 1.1. Let G be a group and X a set. Then an action of G on X is a function F : G × X ! X, where we write F (g; x) = g · x, satisfying: 1. For all g1; g2 2 G and x 2 X, g1 · (g2 · x) = (g1g2) · x. 2. For all x 2 X, 1 · x = x. When the action F is understood, we say that X is a G-set. Note that a group action is not the same thing as a binary structure. In a binary structure, we combine two elements of X to get a third element of X (we combine two apples and get an apple). In a group action, we combine an element of G with an element of X to get an element of X (we combine an apple and an orange and get another orange). Example 1.2. 1. The trivial action: g · x = x for all g 2 G and x 2 X. 2. Sn acts on f1; : : : ; ng via σ · k = σ(k) (here we use the definition of multiplication in Sn as function composition). More generally, if X is any set, SX acts on X, by the same formula: given σ 2 SX and x 2 X, define σ · x = σ(x). To see that this is an action as we have defined it, note that, given σ1; σ2 2 SX and x 2 X, σ1 · (σ2 · x) = σ1 · (σ2(x)) = σ1(σ2(x)) = (σ1 ◦ σ2)(x) = (σ1 ◦ σ2) · x; since the group operation on SX is function composition. Clearly IdX ·x = IdX (x) = x for all x 2 X. Thus SX acts on X. Note that 1 SX acts on many other objects associated to X, such as the power set P(X), the set of all subsets of X, by the formula that, for all σ 2 SX and A ⊆ X, σ · A = σ(A) = fσ(a): a 2 Ag: Since #(σ·A) = #(A), if A is finite, SX also acts on the subset of P(X) consisting of all subsets of X with 2 elements, or with 3 elements, or with k elements for any fixed k. ∗ n 3. The group R acts on the vector space R by scalar multiplication: ∗ n n given t 2 R and v 2 R , let t · v = tv 2 R be scalar multiplication. That this is an action follows from familiar properties of scalar multi- ∗ n plication: t1(t2v) = (t1t2)v and 1v = v, for all t1; t2 2 R and v 2 R . (Of course, these properties hold for t1; t2 2 R as well, but R is not a group under multiplication. Also, scalar multiplication has additional properties having to do with addition of scalars or vectors.) n 4. GLn(R) acts on R by the usual rule A · v = Av is the multiplication of the matrix A on the vector v, and is the same thing as F (v), where n n F : R ! R is the linear function corresponding to A. Similarly, On n and SOn act on R . In addition, On and SOn act on the (n−1)-sphere Sn−1 of radius 1 defined by n−1 n S = fv 2 R : kvk = 1g: 1 2 2 Note that S = U(1) is the unit circle in R and S is the unit sphere 3 in R . An (n − 1)-sphere of radius r is defined similarly. 2 5. Let Pn be a regular n-gon in R , n ≥ 3. For example, we could take Pn to be centered at the origin and to have vertices pk = (cos(2πk=n); sin(2πk=n)); k = 0; 1; : : : ; n − 1: The dihedral group Dn acts Pn and on the set of vertices fp0;:::; pn−1g, as well as on the set of edges fp0p1; p1p2;:::; pn−1p0g. With the above notation, it is easy to see (as we have described in various home- work problems) that Dn = fA 2 O2 : A(Pn) = Png: 6. In a partial analogy with the previous example, let S be a regular solid 3 (or Platonic solid) in R , known to Euclid, Plato, and before Plato to 2 the Pythagoreans. We will not give a precise definition. We view S 2 as centered at the origin. Here, unlike the case of R where there is a regular n-gon for every n ≥ 3, there are just 5 regular solids. A regular solid is an example of a polyhedron, which has vertices, edges and faces. If we tabulate this information, we have the following list of the regular solids (where v is the number of vertices, e is the number of edges, and f is the number of faces): name v e f n tetrahedron 4 6 4 3 cube 8 12 6 4 octahedron 6 12 8 3 dodecahedron 20 30 12 5 icosahedron 12 30 20 3 Here n is the number of edges of a face, which is a regular n-gon. This can be determined by the above data, since each edge meets exactly two faces, and thus 2e = nf. For example, the faces of a dodecahedron are regular pentagons. Note Euler's formula, which in this case says that v − e + f = 2: To every regular solid S there is an associated dual solid S_, where the number of vertices of S_ is equal to the number of faces of S, and vice versa. Here the tetrahedron is its own dual, while the dual of the cube is the octahedron and the dual of the dodecahedron is the icosahedron. Given a regular solid S, we define its symmetry group G(S) by G(S) = fA 2 SO3 : A(S) = Sg: Then G(S) acts on S, and on the sets of vertices, edges, or faces of S. It is not hard to show that G(S) = G(S_). Note that (unlike the case of Dn where we allow elements of O2) we only consider elements of SO3. The group G(S) is always finite, and we shall say a little more about it later. 7. The remaining two examples are more directly connected with group theory. If G is a group, then G acts on itself by left multiplication: g · x = gx. The axioms of a group action just become the fact that multiplication in G is associative (g1(g2x) = (g1g2)x) and the definition 3 of the identity (1x = x for all x 2 G). More generally, if H ≤ G is a subgroup, not necessarily normal, then G acts on the set of left cosets G=H via: g · (xH) = (gx)H. The argument that this is indeed an action is similar to the case of left multiplication. −1 8. G acts on itself by conjugation ig: ig(x) = gxg . (We write it this way instead of as g · x to avoid confusion with the left multiplication action.) To see that this is an action, note that, for all g1; g2 2 G, −1 −1 −1 −1 −1 ig1 (ig2 (x)) = ig1 (g2xg2 ) = g1(g2xg2 )g1 = (g1g2)x(g2 g1 ) −1 = (g1g2)x(g1g2) = ig1g2 (x); −1 −1 −1 where we have used the familiar fact that (g1g2) = g2 g1 . Since clearly −1 i1(x) = 1x1 = 1x1 = x; conjugation does give an action of G on itself. This action is the trivial action () gxg−1 = x for all g; x 2 G () gx = xg for all g; x 2 G () G is abelian. One principle that we have seen implicitly in some of the above examples is the following: Proposition 1.3. If X is a G-set and f : G0 ! G is a homomorphism, then X becomes a G0-set via g0 · x = f(g0) · x. In particular, if H ≤ G, then a G-set X is also an H-set via the inclusion homomorphism of H in G. 0 0 0 Proof. Given g1; g2 2 G , 0 0 0 0 0 0 0 0 g1 · (g2 · x) = g1 · (f(g2) · x) = f(g1) · (f(g2) · x) = ((f(g1)f(g2)) · x 0 0 0 0 = f(g1g2) · x = (g1g2) · x; using the fact that f is a homomorphism. Also, if 10 is the identity in G0, then f(10) = 1 is the identity in G, and thus, for all x 2 X, 10 · x = f(10) · x = 1 · x = x: It follows that the formula g0 · x = f(g0) · x defines an action of G0 on X. Example 1.4. If G is a group, then SG acts on G and on the set P(G) of all subsets of G. Thus, so does Aut G, the group of automorphisms of G (i.e. isomorphisms from G to G), which is a subgroup of G under composition. Note that Aut G also acts on the set of all subgroups of G, which is a subset of P(G), whereas SG does not act on this set (because a bijection from G to itself will not in general take a subgroup to a subgroup). 4 Using Proposition 1.3, we can give a partial generalization of Cayley's theorem. Recall that, for the action of G on itself by left multiplication, we define a bijection `g : G ! G by: `g(x) = gx: More generally, let G act on a set X, and define `g : X ! X by the formula `g(x) = g · x: Lemma 1.5.