This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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)....
View
Full
Document
 Spring '08
 Staff
 Math, Algebra

Click to edit the document details