Orbit-Stabilizer-Proof

Orbit-Stabilizer-Proof - x Therefore | O x | = | G/G x | =...

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Proof from 3/15/10 Theorem 0.1. If G acts on X and x X then | O ( x ) | = [ G : G x ] . Proof. Let G/G x denote the family of cosets of G x in G (not necessarily a quotient group. If y O ( x ) then y = gx for some g G . Let ϕ : O ( x ) G/G x such that ϕ ( y ) = gG x . (Our goal is to show this is a bijection) Well-Defined : If y = hx and y = gx for g,h G then x = h - 1 gx and thus h - 1 g G x hence hG x = gG x . 1-to-1 Suppose ϕ ( y ) = ϕ ( z ). Then y = gx , z = hx and gG x = hG x . Hence h - 1 g G x . Thus h - 1 gx = x which implies gx = hx which in turn implies y = z . onto : If gG x = G/G x let y = gx O ( x ) and ϕ ( y ) = gG
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Unformatted text preview: x . Therefore | O ( x ) | = | G/G x | = [ G : G x ]. ± Question : If y ∈ O ( x ) and g ∈ G x does that imply g ∈ G y ? Answer : No, not necessarily. y ∈ O ( x ) ⇒ α h ( x ) = y for some h ∈ G . g ∈ G x ⇒ α g ( x ) = x . What we know: α h α g α h-1 ( y ) = α h α g ( x ) = α h ( x ) = y . So hgh-1 ∈ G y . 1...
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This note was uploaded on 04/19/2011 for the course MATH 21373 taught by Professor Johnson during the Spring '11 term at Carnegie Mellon.

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