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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

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

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.4: Cyclic Groups
Problem 16E: For an integer n1, let G=Un, the group of units in n that is, the set of all [ a ] in n that have...
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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.
Transcribed Image Text: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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