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SolsFin-113.S11

# SolsFin-113.S11 - Math 113 Spring 2011 Professor Mariusz...

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Math 113, Spring 2011 Professor Mariusz Wodzicki Final Exam (Solutions) May 11, 2011 1. Classify the group G = ( Z × Z )/ (5,6) according to the Fundamental Theorem of Theory of Finitely Generated Abelian Groups . Let π : Z × Z G be the quotient map and ι : (1,1) ⟩ → Z × Z be the inclusion map. We will show that φ = π ι is an isomorphism which means that G is infinite cyclic. Indeed, (1,1),(5,6) contains both (1,0) and (0,1) : (1,0) = 6 · (1,1) - (5,6) and (0,1) = - 5 · (1,1) + (5,6) which generate Z × Z , and thus (1,1),(5,6) ⟩ = Z × Z . It follows that φ is surjective. It is obviously injective, since (1,1) ⟩∩⟨ (5,6) ⟩ = {(0,0)} . 2. Show that the rings 2 Z and 3 Z are not isomorphic. Any ring isomorphism is an isomorphism of the corresponding additive groups. Both 2 Z and 3 Z are infinite cyclic, and an isomorphism between 2 Z and 3 Z must take 2 to a generator of 3 Z , i.e. to 3 or - 3 . In partic- ular, 4 = 2 + 2 would be sent to either 6 = 3 + 3 , or - 6 = - 3 + ( - 3) instead of 9 = ( ± 3) 2 . Thus, none of the two isomorphisms of the additive groups is a homomorphism of multiplicative semigroups. 3. Let α = 1 2 3 4 5 6 7 8 1 5 8 7 6 2 3 4 and β = 1 2 3 4 5 6 7 8 2 5 3 7 6 1 8 4 (1) Find σ S 8 such that σ α = β σ .

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