practice2d

practice2d - MATH 4107 PRACTICE PROBLEMS WITH SOLUTIONS...

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MATH 4107 PRACTICE PROBLEMS WITH SOLUTIONS March 2, 2007 Here are a few practice problems on groups. Try to work these WITHOUT looking at the solutions! After you write your own solution, you can compare to my solution. Your solution does not need to be identical—there are often many ways to solve a problem—but it does need to be CORRECT. 1. Suppose G , A , and B are groups and ψ 1 : G A and ψ 2 : G B are surjective homomorphisms. Suppose, moreover that ker( ψ 1 ) ker( ψ 2 ) = { e } . a. Show that ψ : G A × B , deFned by ψ ( g ) = ( ψ 1 ( g ) , ψ 2 ( g )) is an injective homomor- phism of G into A × B . Solution If g , h G , then, since ψ 1 , ψ 2 are both homomorphisms, ψ ( gh ) = ( ψ 1 ( gh ) , ψ 2 ( gh ) ) = ( ψ 1 ( g ) ψ 1 ( h ) , ψ 2 ( g ) ψ 2 ( h ) ) = ( ψ 1 ( g ) , ψ 2 ( g ) ) ( ψ 1 ( h ) , ψ 2 ( h ) ) = ψ ( g ) ψ ( h ) . Thus ψ is a homomorphism.

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practice2d - MATH 4107 PRACTICE PROBLEMS WITH SOLUTIONS...

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