hw09 - f ( n ) = a n . If a has nite order prove that ker f...

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Modern Algebra 1 Homework 4.2 Spring 2010 Due February 10 Exercise 5. Let f : G H be a group homomorphism. a. Let a G . Prove that f ( a n ) = f ( a ) n for all n Z . b. Let a G . Prove that f ( h a i ) = h f ( a ) i . c. Let a G . Prove that if a has finite order then so does f ( a ). How are their orders related? Justify your answer. d. Let K H . Prove that f - 1 ( K ) G . Exercise 6. Let G be a group, a G and f : Z → h a i be the homomorphism given by
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Unformatted text preview: f ( n ) = a n . If a has nite order prove that ker f = h| a |i . Exercise 7. Let G be an abelian group and let n Z . Dene f : G G by f ( x ) = x n . a. Prove that f is a homomorphism. b. Compute ker f . c. Use f to show that { y G | y = x n for some x G } is a subgroup of G...
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This note was uploaded on 02/29/2012 for the course MATH 3362 taught by Professor Ryandaileda during the Spring '10 term at Trinity University.

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