ACTION OF A GROUP ON A SET

The concept of a group acting on a set X is a slight generalization of the group of symmetries of X. It is equivalent to considering a subgroup of S(X). This concept is useful for determining the order of the symmetry groups of solids in three dimensions, and it is indispensable in Chapter 6, when we look at the Polya´–Burnside method of enumerating sets with symmetries.

The group (G, ·) acts on the set X if there is a function

ψ: G × X → X

such that when we write g(x) for ψ(g, x), we have:

(i)(g1g2)(x) = g1(g2(x)) for all g1, g2 G, x X.

(ii)e(x) = x if e is the identity of G and x X.

Proposition 4.37. If g is an element of a group G acting on the set X, then

Hence g is bijective, and g can be considered as an element of S(X), the group of symmetries of X.

The function χ: G → S(X), which takes the element g G to the bijection g: X → X, is a group morphism because χ(g1g2) is the function from X to X defined by χ(g1g2)(x) = (g1g2)(x) = (g1(g2(x)) = χ(g1) Ž χ(g2)(x); thus χ(g1g2) = χ(g1) Ž χ(g2).

If Kerχ = {e}, then χ is injective, and the group G is said to act faithfully on the set X. G acts faithfully on X if the only element of G, which fixes every

element of X, is the identity e G. In this case, we identify G with Imχ and regard G as a subgroup of S(X).

For example, consider the cyclic group of order 2, C2 = {e, h}, acting on the regular hexagon in Figure 4.4, where h reflects the hexagon about the line joining vertex 3 to vertex 6. Then C2 acts faithfully and can be identified with the subgroup {(1), (15) Ž (24)} of D6.

The cyclic group C3 = {e, g, g2} acts faithfully on the cube in Figure 4.5, where g rotates the cube through one-third of a revolution about a line joining two opposite vertices. This group action can be considered as the subgroup {(1), (163) Ž (457), (136) Ž (475)} of the symmetry group of the cube.

Proposition 4.38. If G acts on a set X and x X, then

Stab x = {g G|g(x) = x}

is a subgroup of G, called the stabilizer of x. It is the set of elements of G that fix x.

Proof. Stab x is a subgroup because

(i)If g1, g2 Stab x, then (g1g2)(x) = g1(g2(x)) = g1(x) = x, so g1g2

Stab x.

The set of all images of an element x X under the action of a group G is called the orbit of x under G and is denoted by

Orb x = {g(x)|g G}.

The orbit of x is the equivalence class of x under the equivalence relation on X in which x is equivalent to y if and only if y = g(x) for some g G. If π is a permutation in Sn, the subgroup generated by π acts on the set {1, 2, . . . , n}, and this definition of orbit agrees with our previous one.

under matrix multiplication. This group is called the special orthogonal group and is isomorphic to the circle group W . SO(2) acts on R2 as follows. The matrix M SO(2) takes the vector x R2 to the vector Mx. The orbit of any element x R2 is the circle through x with center at the origin. Since the origin is the only fixed point for any of the nonidentity transformations, the stabilizer of the origin is the whole group, whereas the stabilizer of any other element is the subgroup consisting of the identity matrix only.

The orbits of the cyclic group C2 acting on the hexagon in Figure 4.4 are {1, 5}, {2, 4}, {3}, and {6}.

There is an important connection between the number of elements in the orbit of a point x and the stabilizer of that point.

Lemma 4.39. If G acts on X, then for each x X

|G: Stab x| = |Orb x|.

Proof. Let H = Stab x and define the function

ξ : G/H → Orb x

by ξ(Hg) = g−1(x). This is well defined on cosets because if Hg = H k, then

k = hg for some h H , so k−1(x) = (hg)−1(x) = g−1h−1(x) = g−1(x), since h−1 H = Stab x.

The function ξ is surjective by the definition of the orbit of x. It is also injective, because ξ(Hg1) = ξ(Hg2) implies that g1−1(x) = g2−1(x), so g2g1−1(x) = x

and g2g1−1 Stab x = H . Therefore, ξ is a bijection, and the result follows.

Note that ξ is not a morphism. G/Stab x is just a set of cosets because Stab x is not necessarily normal. Furthermore, we have placed no group structure on Orb x.

Solution. Let G be the group of proper rotations of a cube; that is, rotations that can be carried out in three dimensions. The stabilizer of the vertex 1 in Figure 4.6 is Stab 1 = {(1), (245) Ž (386), (254) Ž (368)}. The orbit of 1 is the set of all the vertices, because there is an element of G that will take 1 to any other vertex. Therefore, by Theorem 4.40,

2 1

3 4

6 5

7 8

The full symmetry group of the cube would include improper rotations such as the reflection in the plane as shown in Figure 4.7. This induces the permutation (24) Ž (68) on the vertices, and it cannot be obtained by physically rotating the cube in three dimensions. Under this group

Stab 1 = {(1), (245) Ž (368), (254) Ž (386), (24) Ž (68), (25) Ž (38), (45) Ž (36)},

so the order of the full symmetry group of the cube is

|Stab 1||Orb 1| = 6.8 = 48.

Therefore, there are 24 proper and 24 improper rotations of the cube.

The article by Shapiro [32] contains many applications, mainly to group theory, of the actions of a group on a set.

We conclude by mentioning the action of the symmetric group Sn on the set of polynomials in n variables x1, x2, . . . , xn. A permutation σ Sn acts on a polynomial f = f (x1, x2, . . . , xn) by permuting the variables:

σ (f ) = f (xσ 1, xσ 2, . . . , xσ n).

This was the action we used in Chapter 3 to define the parity of a permutation and prove the parity theorem. It has historical interest as well. It is the context in which Lagrange proved Lagrange’s theorem—in essence, what he actually did is prove Theorem 4.40 for this action. Moreover, this is the group action that launched Galois theory, about which we say more in Chapter 11.