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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.
 Fall '08
 EVERAGE
 Algebra, Sets

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