alg-03-16

# alg-03-16 - 3.4. (III-16) GROUP ACTION ON A SET 47 3.4...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

## 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.

### Page1 / 2

alg-03-16 - 3.4. (III-16) GROUP ACTION ON A SET 47 3.4...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online