hw10 - ψ x ψ y for all x,y ∈ G[Putnam Exam 1997 A4...

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Putnam Exam Seminar Assignment 10 Fall 2010 Due November 22 Exercise 1. Let G be a group with identity e and φ : G G a function such that φ ( g 1 ) φ ( g 2 ) φ ( g 3 ) = φ ( h 1 ) φ ( h 2 ) φ ( h 3 ) whenever g 1 g 2 g 3 = e = h 1 h 2 h 3 . Prove that there exists an element a G such that ψ ( x ) = ( x ) is a homomorphism (i.e. ψ ( xy ) =
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Unformatted text preview: ψ ( x ) ψ ( y ) for all x,y ∈ G ). [Putnam Exam, 1997, A4] Exercise 2. Is there a finite abelian group G such that the product of the orders of all its elements is 2 2009 ? [Putnam Exam, 2009, A5]...
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