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College Algebra Exam Review 229

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

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5.1. GROUP ACTIONS ON SETS 239 Definition 5.1.11. Let G act on X . For x 2 X , the stabilizer of x in G is Stab .x/ D f g 2 G W gx D x g . If it is necessary to specify the group, we will write Stab G .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 . Proof. Exercise 5.1.6 . n Proposition 5.1.13. Let G act on X , and let x 2 X . Then W a Stab .x/ 7! ax defines a bijection from G= Stab .x/ onto O .x/ , which satisfies .g.a Stab .x/// D g .a Stab .x// for all g; a 2 G . Proof. Note that a Stab .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.
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