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 : Lagranges Theorem. Given groups H G , then, Ord( H ) | Ord( G ) . Proof. The left-cosets of H form a partition of G . The symbol G means that gp G acts on set ; there is a gp-hom : G S . For g G and , write the gp-action as g ( ) or g ( ) or just g . Define the orbit and stabilizer of a point , and the fixed-pt set of a group-element g : O ( ) : = { g | g G } ; Stab ( ) : = { g G | g = } G ; Fix ( g ) : = { | g = } . This Stab( ) is a subgp, but is rarely normal in G : f G : f Stab( ) f 1 = Stab( f ) . 2 : 3 : Orb-Stab Lemma. For each : Ord ( Stab( ) ) O ( ) = Ord ( G ) . * : Proof. Let H : = Stab( ). Say two elts g , f G are equivalent , g f , if g = f . 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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