feb23 - Math 4124 Wednesday, February 23 February 23,...

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Unformatted text preview: Math 4124 Wednesday, February 23 February 23, Ungraded Homework Exercise 3.3.4 on page 101 Let C be a normal subgroup of the group A and let D be a normal subgroup of the group B . Prove that ( C D ) ( A B ) and ( A B ) / ( C D ) = ( A / C ) ( B / C ) . Define : A B ( A / C ) ( B / D ) by ( a , b ) = ( aC , bD ) . Let a , a 1 A and b , b 1 B . Then ( ( a , b )( a 1 , b 1 ) ) = ( aa 1 , bb 1 ) = ( aa 1 C , bb 1 D ) = ( aCa 1 C , bDb 1 D ) = ( aC , bD )( a 1 C , b 1 D ) = ( a , b ) ( a 1 , b 1 ) Thus is a homomorphism. is onto because given ( aC , bD ) ( A / C ) ( B / D ) , ( a , b ) = ( aC , bD ) . ker = { ( c , d ) | ( c , d ) = e } , which is { ( c , d ) | ( cC , dD ) = ( C , D ) } . Now cC = C if and only if c C , and dD = D if and only if d D . We deduce that ker = C D . The result now follows from the fundamental homomorphism theorem (Theorem 16 on page 97)....
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feb23 - Math 4124 Wednesday, February 23 February 23,...

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