Group Action on a Set - Lecture Notes | MATH 5310, Study notes of Abstract Algebra

Download Group Action on a Set - Lecture Notes | MATH 5310 and more Study notes Abstract Algebra in PDF only on Docsity! 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, 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.