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Homework6Solutions - Homework 6 Solutions Math 332, Spring...

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Homework 6 Solutions Math 332, Spring 2010 Problem 1. (a) Proposition. If G and H are groups, then G × H H × G . Proof. Define ϕ : G × H H × G by ϕ ( g,h ) = ( h,g ). Clearly ϕ is bijective. Moreover, if ( g 1 ,h 1 ) , ( g 2 ,h 2 ) G × H , then ϕ ( ( g 1 ,h 1 )( g 2 ,h 2 ) ) = ϕ ( g 1 g 2 ,h 1 h 2 ) = ( h 1 h 2 ,g 1 g 2 ) = ( h 1 ,g 1 )( h 2 ,g 2 ) = ϕ ( g 1 ,h 1 ) ϕ ( g 2 ,h 2 ) , which proves that ϕ is an isomorphism. (b) Proposition. Let G and H be groups. If A G and B H , then A × B G × H . Proof. We shall use the one-step subgroup test. Clearly A × B is a nonempty subset of G × H . Furthermore, if ( a 1 ,b 1 ) , ( a 2 ,b 2 ) A × B , then a 1 ,a 2 A and b 1 ,b 2 B . Since A and B are subgroups, it follows that a 1 a - 1 2 A and b 1 b - 1 2 B , so ( a 1 ,b 1 )( a 2 ,b 2 ) - 1 = ( a 1 ,b 1 )( a - 1 2 ,b - 1 2 ) = ( a 1 a - 1 2 ,b 1 b - 1 2 ) A × B. (c)
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Homework6Solutions - Homework 6 Solutions Math 332, Spring...

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