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hw25 - y Prove that F is an isomorphism Exercise 4 Let G be...

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Modern Algebra 1 Homework 12.2 Spring 2010 Due April 21 Exercise 3. Let f : G H be a homomorphism of abelian groups. We say that f splits if there is a homomorphism g : H G so that f g = Id H . Show that if f splits then G = ker f × Im f . [ Hint: Define F : ker f × Im f G by F ( x, y ) = x + g ( y
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Unformatted text preview: y ). Prove that F is an isomorphism.] Exercise 4. Let G be a finite abelian group that is not cyclic. Show that there is a prime p that divides | G | so that G contains a subgroup isomorphic to Z p × Z p ....
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