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 : G X X that assigns to each pair ( g,x ) in the Cartesian product set G X an element ( g,x ) = g x in X . If we write g ( x ) = g x we get a mapping g : X X for each g G . We require the action to have the following properties. (1) (a) For each g G , g is a bijection. Hence g Per( X ) , the group of all permutation mappings on X . (b) For each g 1 ,g 2 G we have g 1 g 2 ( x ) = g 1 ( g 2 ( x )) i.e. g 1 g 2 x = g 1 ( g 2 x ) . The latter property is what makes a left action ( covariant action ) on X ; there is a similar definition for right actions , but they wont play a role in our discussion. (c) e ( x ) = x for all x X , so that e = id X . It follows from (1) that each operator g is invertible, and that ( g ) 1 = g 1 , because we have g g 1 = e = id X . square The g are transformations of X . It follows from (1) that the map : G (Per( X ) , ) given by ( g ) = g is a homomorphism from G to the group of permutations on X , in which the product is composition of operators: 1 2 ( x ) = 1 ( 2 ( x )) . The kernel of the homomorphism : G Per( X ) is { g G : 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 g ( x ) = gx (left translation by the element g ). Clearly g 1 g 2 ( x ) = g 1 ( g 2 ( x )) and e ( x ) = x , for all x . Each g is a bijection on X = G because id G = e = g g 1 ; obviously ( g ) 1 = g 1 . This is the action of G on itself by left translations. It is a transitive action . The action kernel is trivial because g ( x ) = gx = x for all x implies (by cancellation)...
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This note was uploaded on 03/25/2011 for the course MATH 301 taught by Professor Algebra during the Spring '11 term at NYU.

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v43-f10trgpsfinal - Notes: F.P. Greenleaf, 2000-2010...

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