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Adv Alegbra HW Solutions 69

Adv Alegbra HW Solutions 69 - 69 Solution Absent(ii Prove...

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Unformatted text preview: 69 Solution. Absent. (ii) Prove that if α is surjective, then α ∗ is surjective. Solution. Absent. (iii) Give an example in which α is injective and α ∗ is not injective. Solution. Absent. 2.104 (i) Prove that Q / Z ( Q ) ∼ = V , where Q is the group of quaternions and V is the four-group. Conclude that the quotient of a nonabelian group by its center can be abelian. Solution. In Exercise 2.86, we saw that Z ( Q ) = {± E } , so that Q / Z ( Q ) has order 4. It is also shown in that exercise that if M = ± I , then M 2 = − I . It follows that every nonidentity element in Q / Z ( Q ) has order 2, and hence, Q / Z ( Q ) ∼ = V (any bijection ϕ : Q / Z ( Q ) → V with ϕ( 1 ) = 1 must be an isomorphism). (ii) Prove that Q has no subgroup isomorphic to V . Conclude that the quotient Q / Z ( Q ) is not isomorphic to a subgroup of Q . Solution. Exercise 2.86 shows that Q has a unique element of order 2, whereas V has 3 elements of order 2....
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