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Unformatted text preview: 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 D 4 of symmetries of a square. Ex 3.55. The symmetric group S n and the alternating group A n of n letters. Ex 3.56. The general linear group GL( n, R ) that contains all nonsingular linear operators in R n . 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. ( g 1 g 2 )( x ) = g 1 ( g 2 x ) for all x X and all g 1 ,g 2 G . When G has an action on X , we call X a G-set . What are the group actions in the preceding examples? Let S X be the group of all permutations of X . A group action ? of G on X a homomorphism of G to S X Thm 3.58. Let G be a group and X be a set....
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This note was uploaded on 10/12/2010 for the course MATH 5310 taught by Professor Staff during the Spring '08 term at Auburn University.
- Spring '08