s09_mthsc851_hw06 - MTHSC 851(Abstract Algebra Dr Matthew...

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MTHSC 851 (Abstract Algebra) Dr. Matthew Macauley HW 6 Due Tuesday March 3nd, 2009 (1) (a) Prove that i I A i is a coproduct in the category of abelian groups. Specifically, let { A i | i I } be a family of abelian groups, and let ι i be the canonical injections, for i I . If B is an abelian group and { f i : A i B | i I } a family of homomorphisms, prove there is a unique homomorphism f : i I A i B such that i = f i for all i I , and this determines i I A i uniquely up to isomorphism. (b) Give an example of how the direct product i I A i fails to be a coproduct in the category of abelian groups. (2) Prove that the free product i I * G i is a coproduct in the category of groups. (3) Let A 1 , A 2 , A be objects in a category C , and let f i Hom( A, A i ) for i = 1 , 2. Suppose that B A 1 g 1 A 2 g 2 A f 1 f 2 and B A 1 g 1 A 2 g 2 A f 1 f 2 are pushouts for ( A, A 1 , A 2 , f 1 , f 2 ). Prove that B and B are equivalent.
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