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v43-f10trgpsfinal

# v43-f10trgpsfinal - Notes F.P Greenleaf 2000-2010...

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Unformatted text preview: Notes: F.P. Greenleaf, 2000-2010 v43-f10trgps.tex, version 11/5/10 Algebra I: Chapter 4. Transformation Groups 4.1 Actions of a Group G on a Space X . Let G be a group and X a set. 4.1.1 Definition. A group action is a map τ : G × X → X that assigns to each pair ( g,x ) in the Cartesian product set G × X an element τ ( g,x ) = g · x in X . If we write τ g ( x ) = g · x we get a mapping τ g : X → X for each g ∈ G . We require the action to have the following properties. (1) (a) For each g ∈ G , τ g is a bijection. Hence τ g ∈ Per( X ) , the group of all permutation mappings on X . (b) For each g 1 ,g 2 ∈ G we have τ g 1 g 2 ( x ) = τ g 1 ( τ g 2 ( x )) – i.e. g 1 g 2 · x = g 1 · ( g 2 · x ) . The latter property is what makes τ a left action ( covariant action ) on X ; there is a similar definition for right actions , but they won’t play a role in our discussion. (c) τ e ( x ) = x for all x ∈ X , so that τ e = id X . It follows from (1) that each operator τ g is invertible, and that ( τ g ) − 1 = τ g − 1 , because we have τ g ◦ τ g − 1 = τ e = id X . square The τ g are “transformations” of X . It follows from (1) that the map Φ : G → (Per( X ) , ◦ ) given by Φ( g ) = τ g is a homomorphism from G to the group of permutations on X , in which the product is composition of operators: σ 1 ◦ σ 2 ( x ) = σ 1 ( σ 2 ( x )) . The kernel of the homomorphism Φ : G → Per( X ) is { g ∈ G : τ g = id X } . It is often referred to as the kernel of the action G × X → X , or simply the action kernel . 4.1.2 Definition. Given a group action G × X → X , each point x ∈ X has a G-orbit G · x = { g · x : g ∈ G } . We say that the action G × X → X is transitive if there is just one orbit: (2) For all x,y ∈ X, there exists a g ∈ G such that g · x = y or equivalently, G · x = X for any x . square In X there is a natural an RST relation R such that (3) x R ∼ y ⇐⇒ ∃ g ∈ G such that y = g · x It is easily verified that R is reflexive, symmetric and transitive. Furthermore, the equiv- alence class of a point x under R is precisely its G-orbit, so [ x ] = { y ∈ X : y R ∼ x } = { y ∈ X : ∃ g ∈ G such that y = g · x } (4) = { g · x : g ∈ G } = ( G-orbit of x ) Thus X splits into disjoint G-orbits which fill X . 4.1.3 Exercise. Verify that the relation R defined in (3) is reflexive, symmetric, and transitive: (i) x R ∼ x , (ii) x R ∼ y ⇒ y R ∼ x , (iii) x R ∼ y and y R ∼ z ⇒ x R ∼ z . square 4.1.4 Examples. Let G be any group. 1 (a) Take X = G and τ g ( x ) = gx (left translation by the element g ). Clearly τ g 1 g 2 ( x ) = τ g 1 ( τ g 2 ( x )) and τ e ( x ) = x , for all x . Each τ g is a bijection on X = G because id G = τ e = τ g ◦ τ g − 1 ; obviously ( τ g ) − 1 = τ g − 1 . This is the action of G on itself by left translations. It is a transitive action . The action kernel is trivial because τ g ( x ) = gx = x for all x implies (by cancellation)...
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v43-f10trgpsfinal - Notes F.P Greenleaf 2000-2010...

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