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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