alg.misc.SLIDES

# alg.misc.SLIDES - Miscellaneous Algebra facts 21 March...

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Unformatted text preview: Miscellaneous Algebra facts 21 March, 2008 (at 18:20 ) Ways to count in groups 1 : Lagrange’s Theorem. Given groups H ⊂ G , then, Ord( H ) •| Ord( G ) . ♦ Proof. The left-cosets of H form a partition of G . The symbol G Ω means that gp G acts on set Ω ; there is a gp-hom ψ : G → S Ω . For g ∈ G and ω ∈ Ω , write the gp-action as ψ g ( ω ) or g ( ω ) or just g ω . Define the orbit and stabilizer of a point ω , and the fixed-pt set of a group-element g : O ψ ( ω ) : = { g ω | g ∈ G } ⊂ Ω ; Stab ψ ( ω ) : = { g ∈ G | g ω = ω } ⊂ G ; Fix ψ ( g ) : = { ω ∈ Ω | g ω = ω } ⊂ Ω . This Stab( ω ) is a subgp, but is rarely normal in G : ∀ f ∈ G : f · Stab( ω ) · f 1 = Stab( f ω ) . 2 : 3 : Orb-Stab Lemma. For each ω ∈ Ω : Ord ( Stab( ω ) ) · O ( ω ) = Ord ( G ) . * : ♦ Proof. Let H : = Stab( ω ). Say two elts g , f ∈ G are “ equivalent ” , g ∼ f , if g ω = f ω . Evidently, the equiv-class of g is simply the left coset gH...
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alg.misc.SLIDES - Miscellaneous Algebra facts 21 March...

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