Do all parts For problems 5-7 find the order of the largest cyclic subgroup of thegiven group.5. Z12 x Z186. Zg x Z20 x Z16(2 Zz x Z10 X Z24For problems 8-10 determine if the groups given are isomorphic.Explain your answer.8. Z4 x Z16 and Zg x Zg9.Z10 x Z70 and Z10 x Z10 x Z,(10.) Z10 x Z70 and Z20 x Z35

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Answered: For problems 5-7 find the order of the…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.6: Quotient Groups

Problem 8E: Suppose G1 and G2 are groups with normal subgroups H1 and H2, respectively, and with G1/H1...

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Chapter 4 Problem 7: Denote G = (Q, +) as the group of rational numbers with addition. Answer the following questions: (1). Prove G is not a cyclic group. (2). Consider A, B as two nontrivial subgroups of G. Is A ∩ B also nontrivial? (3). Prove that for any a,b ∈ G, there exist h ∈ G such that a,b ∈ <x>.
Problem 3. (1) Prove that (ac)(bd) = (abc)(abd) in Sn for distinct a, b, c, d ∈ {1, 2, 3, · · · , n}. (2) Prove that the product of every pair of transpositions in Sn can be expressed as a product of at most two cycles of length 3. (3) Prove that any permutation in the alternating group An with n ≥ 3 is a product of cycles of length 3.

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