In part I. Answer no. 4 ,and part II answer no. 1 and 2 I. Provide a two-column proof to the following statements.1. Let G be a group with the property that for any x, y,z in the group, xy = zximplies y = z. Prove that G is Abelian.2. Prove that a group G is Abelian if and only if (gh)-1 = g-1h-1 for all g,h e G.3. Prove that in a group, (g-1)-1 = g for all g.4. Prove that in a group, (ab)? = a²b² if and only if ab = ba.II. Provide a complete solution to the following.1. Let a, b, and c be elements of a group.(i) Solve the equation axb = c for x.(ii) Solve a-1xa = c for x.2. In Zg = {0,1, 2, 3, 4} under multiplication modulo 5, what elements satisfy theequation x5 = 1.

Answered: Image /qna-images/answer/5abbade4-942b-4cd2-838e-a08c24336a7b.jpg

Answered: implies y = z. Prove that G is Abelian.…

implies y = z. Prove that G is Abelian. Prove that a group G is Abelian if and only if (gh)-1 = g¬ Prove that in a group, (g-1)-1 = g for all g. %3D

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

In part I. Answer no. 4 ,and part II answer no. 1 and 2

I. Provide a two-column proof to the following statements.
1. Let G be a group with the property that for any x, y,z in the group, xy = zx
implies y = z. Prove that G is Abelian.
2. Prove that a group G is Abelian if and only if (gh)-1 = g-1h-1 for all g,h e G.
3. Prove that in a group, (g-1)-1 = g for all g.
4. Prove that in a group, (ab)? = a²b² if and only if ab = ba.
II. Provide a complete solution to the following.
1. Let a, b, and c be elements of a group.
(i) Solve the equation axb = c for x.
(ii) Solve a-1xa = c for x.
2. In Zg = {0,1, 2, 3, 4} under multiplication modulo 5, what elements satisfy the
equation x5 = 1.

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