Problem 3. In this problem, we will write [b] for the element [b] € Z/kZ.3.1. Show that the map: Z/mnZ(Z/mZ) × (Z/nZ), 4([a]mn) = ([a]m, [a]n)is well-defined, and is a group homomorphism.3.2. Now suppose in addition that m and n are relatively prime. Determine ifis an isomorphism.3.3. If m and n are not relatively prime, can be an isomorphism?4

1) 2) When m and n are relatively prime, by the Chinese Remainder Theorem, there is a bijective…

Answered: Problem 3. In this problem, we will…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.CM: Cumulative Review

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Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).
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10.3.3. Let G = Z/4Z × Z/6Z, and let H be the subgroup of G generated by (2,2). (a) What are the elements of H? (b) What are the elements of G/H? (c) Find a familiar group that is isomorphic to G/H.
7. [10] Let G be a group, with x E G and o (x) = 48. Give the order of the following elements of G: a) o(x²) b) o (x20) c) o(x23)
6. Show that for any two elements x, y of any group G, o(xy) = o(yx). %3D
(4.5) #2 Modern Applied Algebra. Show work.
3. Let Dn (n ≥ 3) be the dihedral group of order 2n. (i) Show that D10 ≈ D5 × Z2 by constructing an explicit isomorphism between the two groups. (ii) What are the centers of D5 and D10? (iii) Identify the quotient groups D5/Z(D5) and D10/Z(D₁0) in terms of known groups.
1.15. Write out the following tables for Z/mZ and (Z/mZ)*, as we did in Figur and 1.5. Exercises Fit to page (a) Make addition and multiplication tables for Z/3Z. (b) Make addition and multiplication tables for Z/6Z. (c) Make a multiplication table for the unit group (Z/9Z)*. (d) Make a multiplication table for the unit group (Z/16Z)*.
6.14. Let π E SÅ be the permutation defined by TT (2) = 8, T(1) = 3, π(5) = 6, TT (6) = 1, TT (3) = 5, π(7) = 7, T(4) = 4, π(8) = 2. (a) Prove that ☛ has order 4. Let G be the subgroup for S8 generated by ; i.e., G {е, π, T², T³}. = = (π) = (b) Describe all of the orbits of G, as was done in Example 6.18. (c) Let X {1, 2, 3, 4, 5, 6, 7, 8}, so G acts on X. For each k € X, describe the stabilizer Gk. (d) Use your data from (b) and (c) to explicitly verify the orbit-stabilizer counting formula in Theorem 6.21. See (6.9) in Example 6.22 for a similar example.

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