Quiz3ProblemSolutions

Quiz3ProblemSolutions - Quiz 3 Practice Problem Solutions...

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Quiz 3 Practice Problem Solutions Math 332, Spring 2010 Questions on Groups 1. The group G/N has four elements: N = { 1 , 4 } , 2 N = { 2 , 8 } , 7 N = { 7 , 13 } , 11 N = { 11 , 14 } . Since (2 N ) 2 = (7 N ) 2 = (11 N ) 2 = N , the isomorphism type is Z 2 × Z 2 (or V ). 2. Since ( (1 2 3 4) N ) 2 = (1 3)(2 4) N = N , the given element has order 2 . 3. Since s N and r - 1 sr = r 6 s 3 N , the subgroup N is not normal . 4. If aN,bN G/N , then ( aN )( bN ) = ( ab ) N = ( ba ) N = ( bN )( aN ), which proves that G/N is abelian. 5. { (0 , 0) , (1 , 1) , (2 , 2) } . 6. If ( g 1 ,h 1 ) , ( g 2 ,h 2 ) G × H , then π ( ( g 1 ,h 1 )( g 2 ,h 2 ) ) = π ( g 1 g 2 ,h 1 h 2 ) = g 1 g 2 = π ( g 1 ,h 1 ) π ( g 2 ,h 2 , which proves that π is a homomorphism. 7. Clearly S is a subset of G , and S is nonempty since e S . Moreover, if g 1 ,g 2 S , then ϕ ( g 1 ) = g 1 and ϕ ( g 2 ) = g 2 , so ϕ ( g 1 g - 1 2 ) = ϕ ( g 1 ) ϕ ( g 2 ) - 1 = g 1 g - 1 2 . It follows that g 1 g - 1 2
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Quiz3ProblemSolutions - Quiz 3 Practice Problem Solutions...

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