Contemporary Abstract Algebra

9th Edition

ISBN: 9781305657960

Author: Joseph Gallian

Publisher: Cengage Learning

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Textbook Question

Chapter 6, Problem 6E

Prove that isomorphism is an equivalence relation. That is, for anygroups G, H, and K
G ≈ G ; G ≈ H implies H ≈ G G ≈ H and H ≈ K implies G ≈ K .

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Chapter 6, Problem 5E

Chapter 6, Problem 7E

Chapter 6 Solutions

Contemporary Abstract Algebra

Ch. 6 - Prob. 1E Ch. 6 - Find Aut(Z).Ch. 6 - Let R+ be the group of positive real numbers under...Ch. 6 - Show that U(8) is not isomorphic to U(10).Ch. 6 - Show that U(8) is isomorphic to U(12).Ch. 6 - Prove that isomorphism is an equivalence relation....Ch. 6 - Prove that S4 is not isomorphic to D12 .Ch. 6 - Show that the mapping alog10a is an isomorphism...Ch. 6 - In the notation of Theorem 6.1, prove that Te is...Ch. 6 - Given that is a isomorphism from a group G under...

Ch. 6 - Let G be a group under multiplication, G be a...Ch. 6 - Let G be a group. Prove that the mapping (g)=g1...Ch. 6 - Prob. 13E Ch. 6 - Find two groups G and H such that GH , but...Ch. 6 - Prob. 15E Ch. 6 - Find Aut(Z6) .Ch. 6 - If G is a group, prove that Aut(G) and Inn(G) are...Ch. 6 - If a group G is isomorphic to H, prove that Aut(G)...Ch. 6 - Suppose belongs to Aut(Zn) and a is relatively...Ch. 6 - Let H be the subgroup of all rotations in Dn and...Ch. 6 - Let H=S5(1)=1andK=S5(2)=2 . Provethat H is...Ch. 6 - Show that Z has infinitely many subgroups...Ch. 6 - Prob. 23E Ch. 6 - Let be an automorphism of a group G. Prove that...Ch. 6 - Prob. 25E Ch. 6 - Suppose that :Z20Z20 is an automorphismand (5)=5 ....Ch. 6 - Identify a group G that has subgroups isomorphic...Ch. 6 - Prove that the mapping from U(16) to itself given...Ch. 6 - Let rU(n) . Prove that the mapping a: ZnZn defined...Ch. 6 - The group {[1a01]|aZ} is isomorphic to what...Ch. 6 - If and are isomorphisms from the cyclic group a...Ch. 6 - Prob. 32E Ch. 6 - Prove property 1 of Theorem 6.3. Theorem 6.3...Ch. 6 - Prove property 4 of Theorem 6.3. Theorem 6.3...Ch. 6 - Referring to Theorem 6.1, prove that Tg is indeed...Ch. 6 - Prove or disprove that U(20) and U(24) are...Ch. 6 - Show that the mapping (a+bi)=a=bi is an...Ch. 6 - Let G={a+b2a,barerational} and...Ch. 6 - Prob. 39E Ch. 6 - Explain why S8 contains subgroups isomorphic to...Ch. 6 - Let C be the complex numbers and M={[abba]|a,bR} ....Ch. 6 - Prob. 42E Ch. 6 - Prob. 43E Ch. 6 - Suppose that G is a finite Abelian group and G has...Ch. 6 - Prob. 45E Ch. 6 - Prob. 46E Ch. 6 - Suppose that g and h induce the same inner...Ch. 6 - Prob. 48E Ch. 6 - Prob. 49E Ch. 6 - Prob. 50E Ch. 6 - Prob. 51E Ch. 6 - Let G be a group. Complete the following...Ch. 6 - Suppose that G is an Abelian group and is an...Ch. 6 - Let be an automorphismof D8 . What are the...Ch. 6 - Let be an automorphism of C*, the group of...Ch. 6 - Let G=0,2,4,6,...andH=0,3,6,9,... .Prove that G...Ch. 6 - Give three examples of groups of order 120, no two...Ch. 6 - Let be an automorphism of D4 such that (H)=D ....Ch. 6 - Prob. 59E Ch. 6 - Prob. 60E Ch. 6 - Write the permutation corresponding to R90 in the...Ch. 6 - Show that every automorphism of the rational...Ch. 6 - Prove that Q+ , the group of positive rational...Ch. 6 - Prob. 64E Ch. 6 - Prob. 65E Ch. 6 - Prove that Q*, the group of nonzero rational...Ch. 6 - Give a group theoretic proof that Q under addition...

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Prove that isomorphism is an equivalence relation. That is, for anygroups G , H , and K G ≈ G ; G ≈ H implies H ≈ G G ≈ H and ​ H ≈ K implies G ≈ K .

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Chapter 6 Solutions

Prove that isomorphism is an equivalence relation. That is, for anygroups G , H , and K G ≈ G ; G ≈ H implies H ≈ G G ≈ H and H ≈ K implies G ≈ K .