Group Actions

In this section we introduce the concept of an action of a group G on a non-empty set X and study such actions. We show that a group action of G on X assigns to each element of G a permutation on the set X. This leads to a proof of Cayley’s theorem that every abstract group is isomorphic to some group of permutations and, in particular, every group of order n is isomorphic to a certain subgroup of S_n.

Definition 7.17.1 An action of the group G on a non-empty set X is a function

such that

1. e * x = x (e is the identity of G), for any x ∈ X;

2. (g₁g₂) * x = g₁ * (g₂ * x), for any x ∈ X, g₁, g₂ ∈ G .

The notion of a group action is equivalent to the notion of a homomorphism from a group to a symmetric group, in the following sense: if ψ is a homomorphism from a group G to a symmetric group S(X), then for every element g ∈ G there is a bijections ψ_g from X to X given by ψ_g(x) = g * x, with inverse ψ_g⁻¹, which defines an action of G on X. Conditions i) and ii) state that the mapping ψ from G to the group S(X) made by of all bijections (symmetric group) from X to itself is given by

a group homomorphism called a permutation representation of G on X. Then there is a corresponding left action of G on X defined by

Conversely, if we start with a left action of G on X, say (g, x) ⟼ g * x, there is a corresponding permutation representation ψ: G → Sym(X) defined by ψ_g(x) = g * x, where g ∈ G, x ∈ X. Again it is an easy verification that the mapping σ is a homomorphism and hence is a permutation representation of G on X.

Thus the homomorphism ψ represents elements of the abstract group G by concrete objects, namely permutations of X. A permutation representation provides a useful way of visualizing the elements of an abstract group.

We say also that the group G acts on the set X (as transformation group) which is called G-set.

If ψ is a permutation representation of a group G on a set X, then G/ Ker(ψ) ≃ Im(ψ) by the First Isomorphism Theorem 7.13.2. Thus G/ Ker(ψ) is isomorphic with a permutation group on X. If Ker(ψ) = e, then G itself is isomorphic with a permutation group on X, in which case the representation ψ is said to be faithful. The term faithful can also be applied to a group action by means of the associated permutation representation.

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 𝒪(x):

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

This is an equivalence relation. Equivalence classes are called orbits. For every x ∈ X and g ∈ G we shall indicate, g(x) or g * x simply as gx: pay attention to the fact that this shorthand has not the meaning of a multiplication between elements!

In terms of movements, the orbit of an element x ∈ X is made by all y ∈ X in which the element x can be moved by the elements of G.

Definition 7.17.2 (Stabilizer). The stabilizer of an element x in G is the set of group elements g that don't move x that is the set of elements of G that fix x. It is denoted as Stab(x).

Proposition 7.17.3 If G acts on a set X and x ∈ X then Stab (x) is a subgroup of G.

Proof. Stab x is a subgroup because

1. If g₁, g₂ ∈ Stab (x), then (g₁g₂)(x) = g₁(g₂(x)) = g₁(x) = x, so g₁g₂ ∈ Stab x.

2. If g ∈ Stab x, then g⁻¹(x) = x, so g⁻¹ ∈ Stab x. □

Definition 7.17.4. The centralizer, C_a of a ∈ G is its stabilizer under conjugation (see Example 7.17.58a). Thus

It consists of those elements in G which commute with a. The centralizer always contains the group center of the group: Z(G) ⊆ C(a) for all a ∈ G; in fact

Clearly a ∈ Z(G) ⇐⇒  C(a) = G; In an Abelian group, the centralizer is the whole group. The centralizer is contained in the corresponding normalizer.

Proposition 7.17.5. The subgroup St_x of G. In addition

that is the stabilizers of elements which are in the same orbit are conjugates.

Proof. For the second part of the proposition. Let w = gx then St_w := {y ∈ G | yw = w}. Then if y ∈ St_gx yw = w or y(gx) = gx.

Every group acts on itself, by left and right multiplication, and by conjugation.

Examples 7.17.6

1. Action by Conjugation. Conjugation is an important special case of an action of a group on a set. Every group G acts on itself, that is on X = G by conjugation. In this case

   g * x := gxg⁻¹

   The orbit of an element x ∈X (=G) is made by all elements y ∈ G (=X) such that y = gxg⁻¹ for some g ∈ G. The orbits are this conjugate classes. From here it appears clear why the writing g * x does not represent a product gx in G (which would have sense), but the conjugate element gxg⁻¹.

   The stabilizer St_x of an element x ∈ X is given by

   St_x = {g ∈ G | gxg⁻¹ = x} = {g ∈ G | gx = xg}

   it coincides with the centralizer of the element x, C(x).

2. The symmetric group S_n acts naturally on the set X = {1,2,...,n}

   σ * x = σ(x) = the trnasformed through σ of x.

   The orbit of an element x ∈ X is made by all those element y ∈ X such for which there exist a σ ∈ S_n such that y = σ(x). Given an element x ∈ X there will certainly be an element y ∈ X because there is a permutation that sends x in y

3. Let σ = (4,5) (1,3,6) (2,7,8) ∈ S₈ and let G = ⟨σ⟩. The the orbits under which X = {1,2,...,8} is parted under the action of G on X are three: {4,5}, {1,3,6} and {2,7,8}. The group G here is nothing but all possible group elements σⁿ for non-negative integers n. Remember that in any finite cyclic group, such as this , all negative powers can be rewritten as positive powers. In this case, that sigma has order 6 (as it turns out), so σ⁻¹ = σ^{(6 − 1)} = σ⁵, σ⁻² = σ^{(6 − 2)} = σ⁴, etc. So, more generally, for any x of finite order n in any group at all, x⁻¹ is the same thing as x⁽ⁿ⁻¹⁾ So for example: σ1 = 3, σσ1 = σ3 = 6, σσσ1 = σσ3 = σ6 = 1. So that orbit is done and we have used up 1,3,6 from X so far.Then then the cycle diagram of σ is a graphical depiction of the action of G on X.

4. Let G = GL_n(ℝ) the linear group made by all invertible matrices with real elements. This group acts on X = ℝⁿ by the following natural action

   A * x : = Ax

   where x (x₁, x₂, ..., x_n)^T and Ax is the ordinary matrix product.

5. Let H a subgroup of a group G. We define the following action of the group H on G:

   h * g := hg   (multiplication in G) ∀ h ∈ H, ∀g ∈ G

   This action corresponds with the ordinary group multiplication in G. The orbits are the right cosets modulo the subgroup H. The action axiom is verified since

   (h₁h₂) g = h₁(h₂g)   ∀g ∈ G, ∀h₁,h₂ ∈ H

We remark that if the stabilizer of an element x ∈ X is large then its orbit is small.

Example: Cayley’s Theorem. We know that any group G acts on the set X = G by left multiplication. The preceding construction produces a group homomorphism φ: G → Sym(G) such that φ(g) = L_g for all g ∈ G, and L_g(x) = gx for all x ∈ G. We claim that φ is injective in this situation. For, suppose g, h ∈ G and L_g = L_h. Applying these two functions to e (the identity of G) gives L_g(e) = L_h(e), so ge = he, so g = h. It follows that G is isomorphic (via φ) to the image of φ, which is a subgroup of the symmetric group Sym(G). This proves Cayley’s Theorem, which says that any group is isomorphic to a subgroup of some symmetric group. If G has n elements, one can check that Sym(G) is isomorphic to S_n. So every n-element group is isomorphic to a subgroup of the symmetric group S_n.  ■

Proposition 7.17.7. Let X a G-set. The cardinality of the orbit 𝒪(x) of the element x ∈ X is equal to the index St_x in G

Proof. It suffices to find a bijections between 𝒪(x) and right cosets

We let

It is a well-defines application and injective. Indeed

Note on the last iff: Apply the definition of right coset: a ~_r b ⇐⇒ ab⁻¹ ∈ St_x which in this case is: g₂⁻¹ ~_r g₁⁻¹.

In addition the map is surjective since every right coset St_x g has the counterimage g⁻¹x ∈ O(x).   □

Corollary 7.17.8. The cardinality of the conjugacy class of an element x of a group G is equal to the index of the centralizer C(x).

Corollary 7.17.9. If G is a finite group acting on a set X, for every x ∈ X we have

Example 7.17.10. Let G a group which acts on a subgroup H of G, where the action is the multiplication like in example 7.17.5(e). Let g ∈ G. Then O(g = Hg, St_g = {e}, thus the index of St_g in H is equal to |H| as dictated by Proposition 7.17.6.

Corollary 7.17.11. Consider the natural action of S₄ on X = {1,2,3,4}. Take x = 1 ∈ X. Then

thus |𝒪(1)| = 4, |St₁| = 6 then 4 ⋅ 6 = 24 = |S₄|   ■.

Example 7.17.12. Let X the set of words made by seven letters. We want to count how many distinct words with two As, three Bs and on two Cs are possible. The group S₇ acts on X permuting the letters: for example,

We calculate to this end the stabilizer of x: St_x is made by all permutations σ ∈ S₇ such that

(135) moves the 1st letter to the 3rd spot, the 3rd letter to the 5th spot, and the 5th letter to the 1st spot. In total the stabilizer has 2!3!2! = 24 elements. Then

The Class equation

In case the action is the coniugacy action we have the following result:

Corollary 7.17.13. Let G a finite group. Let C(x) the centraliser of the element x ∈ G, then since the conjugacy classes partition G, we've the following relation

then

where the sum is extended to the x ∈ G, one for every conjugacy class.

Proof. Let c_x the number of conjugate elements of x ∈ G. A finite group G equals the number of conjugacy classes in G

where an x is considered for every conjugacy class. Since c_x = |O(x)| and C(x) = St_x, the relation follows from corollary 7.17.6.   □

Equation 7.17.1 is known as class equation.

Remark. Recalling the definition of center Z(G) of a group and noticing that an element x belongs to the center iff its conjugacy class is made by the only element x:

The set of elements z whose conjugacy class consists of z alone is precisely the center of the group i.e. Z(G is a union of one-element conjugacy classes), 7.17.1 can be written as

where the sum is over x ∉ Z(G), one for every conjugacy class.

Example 7.17.14. Let G = S₃. The conjugacy classes are

Indicating with C(x) the centralizer of the element x, taking an x in every conjugacy class

Verifyng the class equation

7.17.14 Burnside's Theorem. Let G a finite group and let X a finite G-set. Then the number, s of orbits in X with respect to the action of G is given as follows

where |X_g| indicates the set

Proof. Let N the cardinality of the set made by all couple (g, x) such that gx = x. Fixing g ∈ G, there exist |X_g| couples having g as first element. Fixing instead x, there are |St_x| couples having x as second element. Thus we have the following equality

By corollary 7.17.8, the right side of the previous relation is equal to

Since ∑_{x ∈ X} 1/|𝒪(x)| represents exactly the number s of orbits we are looking for. Eq 7.17.2 can thus be written as

and the theorem is thus proved.  □

Example 7.17.15. Let us use the Burnside counting theorem to find the number of ways of seating 10 people around a circular table. Let X the set of all possible ordering of the 10 people. We have |X| = 10!, which corresponds to the number of ways to seat the people in a row. A very obvious difference between seating people on a straight bench and seating them at a circular table is that each person has 2 neighbours in the latter case and as long as each person retains the same 2 neighbours, they can be rotated round the table without causing any difference. Hence the 10 arrangements in a row correspond to one arrangement in a circle.

The number of ways to arrange 10 men at a round table is 10!/10 = 9!

Example 7.17.16. How many distinct bracelets can be made with five red beads and three blue beads?

Every configuration is determined once we fix three beads, thus all the configurations are C_8,3 = 56. For each combination by rotating the pearls in the bracelet we consider it to be the same braeclet D₈ because you can flip the bracelet over and its the same one. Rotation means that you rotate the beads around the bracelet respecting the same ordering of beads. If x = RRRRRBBB when is rotated becomes y = RRRRBBBR. So rotations do not leave any element fixed. Reflections through the axes of the polygon do not fix any configuration whilst reflections along the diagonal of the polygon fix 6 configurations (those having a red and blue bead on the vertex of the polygon)