It is relevant in number theory to understand the group structure of Z. In this problem we will work out the structure of Z when n = 2m. The cases m = 1 and 2 are easy to compute directly: we have Z {1} is the trivial group, and Z₁ = {1,3} Z₂. So we suppose m≥ 3, and let G = Z₂m. m = = (a) Use mathematical induction to show that for every integer k ≥ 0, we have 52 = 1+2k+2q for some odd integer q. (b) Show that the order of 5 in G is 2m-2

It is relevant in number theory to understand the group structure of Z. In this problem we will work out the structure of Z when n = 2m. The cases m = 1 and 2 are easy to compute directly: we have Z {1} is the trivial group, and Z₁ = {1,3} Z₂. So we suppose m≥ 3, and let G = Z₂m. m = = (a) Use mathematical induction to show that for every integer k ≥ 0, we have 52 = 1+2k+2q for some odd integer q. (b) Show that the order of 5 in G is 2m-2

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.4: Cyclic Groups

Problem 10E: Exercises 10. For each of the following values of, find all subgroups of the cyclic group under...

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