hw8 - (2 points) 4. Problem 26.14 on page 131 (to show ( a...

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Math 3124 Thursday, November 3 Eighth Homework Due 12:30 p.m., Tuesday November 15 1. Let G = GL 2 ( Z 5 ) and define θ : G ( Z # 5 , ± ) by θ ( A ) = det ( A ) . (a) Prove that θ is a homomorphism of G onto Z # 5 . You may assume the multiplicative property for determinants of matrices over Z n , namely det ( AB ) = det ( A ) ± det ( B ) . (b) Determine ker θ . (c) If K = { A G | det ( A ) = [ 1 ] } , prove that K ± G and that G / K = Z # 5 . (d) Is G / K cyclic? (3 points) 2. Let G = ( Z # 509 , ± ) and let H = { x 4 | x G } . Determine | H | . Help: [ 208 ] 2 = [ - 1 ] . (2 points) 3. Problem 24.17 on page 125
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Unformatted text preview: (2 points) 4. Problem 26.14 on page 131 (to show ( a + b 2 )-1 Q [ 2 ] , multiply top and bottom by a-b 2). (2 points) 5. Let R be a commutative ring of characteristic 3. Prove that if a , b R , then ( a + b ) 3 = a 3 + b 3 . (In other words, prove the Freshman calculus rule for the case p = 3.) (2 points) (5 problems, 11 points altogether)...
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This note was uploaded on 01/02/2012 for the course MATH 3124 taught by Professor Parry during the Fall '08 term at Virginia Tech.

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