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Unformatted text preview: MODEL ANSWERS TO THE SEVENTH HOMEWORK 1. For Chapter 2, Section 9: 1. Let φ : G 1 × G 2→ G 2 × G 1 be the homomorphism that sends ( g 1 , g 2 ) to ( g 2 , g 1 ). This is clearly a bijection. We check that it is a homomorphism. Suppose that ( g 1 , g 2 ) and ( h 1 , h 2 ) ∈ G 1 × G 2 . Then φ (( g 1 , g 2 )( h 1 , h 2 )) = φ ( g 1 h 1 , g 2 h 2 ) = ( g 2 h 2 , g 1 h 1 ) = ( g 2 , g 1 )( h 2 , h 1 ) = φ ( g 1 , g 2 ) φ ( h 1 , h 2 ) . Thus φ is a homorphism. Alternatively, we could use the universal property of the product. Both G 1 × G 2 and G 2 × G 1 satisfy the universal properties of a product and so they must be isomorphic, by uniqueness. 1. For Chapter 2, Section 9: 2. These properties are clearly preserved by isomorphism, so we may as well assume that G 1 = Z m and G 2 Z n . Consider (1 , 1) ∈ G 1 × G 2 . Suppose that k (1 , 1) = (0 , 0). Then k = 0 mod m and k = 0 mod n . As m and n are coprime it follows that k = mod mn . But then the order of (1 , 1) is at least mn . As G 1 × G 2 is a group of order mn , it follows that G 1 × G 2 is cyclic, generated by (1 , 1)....
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 Spring '08
 LONG
 Math, Algebra, Vector Space, Category theory, Cyclic group, Homomorphism

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