College Algebra Exam Review 229

College Algebra Exam Review 229 - 239 5.1. GROUP ACTIONS ON...

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Unformatted text preview: 239 5.1. GROUP ACTIONS ON SETS Definition 5.1.11. Let G act on X . For x 2 X , the stabilizer of x in G is Stab.x/ D fg 2 G W gx D x g. If it is necessary to specify the group, we will write StabG .x/. Lemma 5.1.12. For any action of a group G on a set X and any x 2 X , Stab.x/ is a subgroup of G . I Proof. Exercise 5.1.6. Proposition 5.1.13. Let G act on X , and let x 2 X . Then W aStab.x/ 7! ax defines a bijection from G=Stab.x/ onto O .x/, which satisfies .g.aStab.x/// D g .aStab.x// for all g; a 2 G . Proof. Note that aStab.x/ D b Stab.x/ , b 1 a 2 Stab.x/ , b 1 ax D x , bx D ax . This calculation shows that is well defined and injective. If y 2 O .x/, then there exists a 2 G such that ax D y , so .aStab.x// D y ; thus is surjective as well. The relation .g.aStab.x/// D g .aStab.x// for all g; a 2 G is evident from the definition of . I Corollary 5.1.14. Suppose G is finite. Then jO .x/j D ŒG W Stab.x/ D jG j : jStab.x/j In particular, jO .x/j divides jG j. Proof. This follows immediately from Proposition 5.1.13 and Lagrange’s theorem. I ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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