hw26 - f p of the preceding exercise is not onto. Conclude...

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Modern Algebra 1 Homework 12.3 Spring 2010 Due April 21 Exercise 5. Let G be an (additive) abelian group and let m Z . a. Prove that the function f m : G G given by f m ( x ) = mx is a homomorphism. b. Use part a to show that G m and mG = { mx | x G } are both subgroups of G . c. If G is finite, under what conditions on m and | G | is f m injective? Exercise 6. If p is a prime and G is an abelian p -group, show that the homomorphism
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Unformatted text preview: f p of the preceding exercise is not onto. Conclude that | pG | < | G | . Exercise 7. If p is a prime and n N prove that p Z p n = Z p n-1 . [ Suggestion: Use the homomorphism f p of exercise 5 and the fact that subgroups of cyclic groups are cyclic.]...
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