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Quiz3Problems - Quiz 3 Practice Problems Math 332 Spring...

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Quiz 3 Practice Problems Math 332, Spring 2010 Questions on Groups 1. Let G = U (15), and let N = { 1 , 4 } . Determine the isomorphism type of G/N . 2. Let G = S 4 , and let N = { e, (1 2)(3 4) , (1 3)(2 4) , (1 4)(2 3) } . Determine the order of the element (1 2 3 4) N in G/N . 3. Let G = D 8 , and let N = { e, s, r 4 , r 4 s } . Is N is a normal subgroup of D 8 ? Explain. 4. Let G be an abelian group, and let N be a normal subgroup of G . Prove that G/N is abelian. 5. Let ϕ : Z 3 × Z 3 Z 3 be a homomorphism, and suppose that ϕ (1 , 0) = 1 and ϕ (0 , 1) = 2. List the elements in the kernel of ϕ . 6. Let G and H be groups, and define a function π : G × H G by π ( g, h ) = g. Prove that π is a homomorphism. 7. Let G be a group, let ϕ : G G be a homomorphism, and let S = { g G | ϕ ( g ) = g } . Prove that S G . 8. List all isomorphism types of abelian groups of order 900. 9. List seven non-isomorphic abelian groups of order 32. 10. Determine whether the following pairs of groups are isomorphic: (a) Z 12 × Z 30 and Z 3 × Z 6 × Z 20 .
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