3.4. (III-16) GROUP ACTION ON A SET 47

3.4 (III-16) Group Action on a Set

3.4.1 Group Action We have seen many examples of group acting on a set.

Ex 3.54. The group D4 of symmetries of a square.

Ex 3.55. The symmetric group Sn and the alternating group An of n letters. Ex 3.56. The general linear group GL(n, R) that contains all nonsingular linear operators in Rn.

Given a map ? : G × X → X, we denote gx := ?(g, x) ∈ X. Note that gx is NOT a group multiplication since g ∈ G and x ∈ X.

Def 3.57. Let G be a group and X be a set. An action of G on X is a map ? : G × X → X such that

1. ex = x for all x ∈ X.

2. (g1g2)(x) = g1(g2x) for all x ∈ X and all g1, g2 ∈ G. When G has an action on X, we call X a G-set.

What are the group actions in the preceding examples? Let SX be the group of all permutations of X. A group action ? of G on X ⇐⇒ a homomorphism φ of G to SX Thm 3.58. Let G be a group and X be a set.

1. If ? : G × X → X is a group action, then φ : G → SX , defined by φ(g) := ig such that ig(x) := gx, is a homomorphism from G to SX .

2. If φ : G → SX is a homomorphism. Then ? : G × X → X, defined by gx := φ(g)(x), is a group action of G on X.

Proof.

In the preceding theorem,

0 ker(φ) = {g ∈ G | φ(g) = e ∈ SX } = {g ∈ G | gx = x for x ∈ X} is a normal subgroup of G. When ker(φ) = {e}, we say that G acts faithfully on X. 48 CHAPTER 3. HOMOMORPHISMS AND FACTOR GROUPS

Def 3.59. Let X be a G-set. If e ∈ G is the only group element that sends every x to x, then we say that G acts faithfully on X. A group G acts transitive on X, if for each x1, x2 ∈ X, there exists g ∈ G such that gx1 = x2. Ex 3.60 (Ex 16.4, p.156). Every group G is itself a G-set

Ex 3.61 (Ex 16.5, p.156). Let H be a subgroup of G. Then G is a H-set under conjugation where ?(h, g) := hgh−1 for h ∈ H and g ∈ G.

3.4.2 Isotropy Groups Let X be a G-set. Let x ∈ X and g ∈ G. Denote

Xg := {x ∈ X | gx = x},Gx := {g ∈ G | gx = x}.

Then Xg is a subset of X, and Gx turns out to be a subgroup of G.

Thm 3.62. Let X be a G-set. Then Gx is a subgroup of G for each x ∈ X. Called Gx the isotropy subgroup of x.

3.4.3 Orbits Def 3.63. Let X be a G-set. Let x ∈ X. Then Gx := {gx | g ∈ G} is the orbit of x under G.

When X is a G-set, we can partition X into disjoint unions of G orbits.

Thm 3.64. Let X be a G-set and x ∈ X. Then |Gx| = (G : Gx).

Ex 3.65. Let σ := (1, 2, 5, 7)(3, 6) ∈ S8. What is the orbit of the letter 2? In this example, indicate G, X, x, Gx, Gx, and Xσ. Then verify that |Gx| = (G : Gx).

3.4.4 Homework, Sect 16, p.159-161 8, 11, 12, 18.