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(3) Suppose n= |T(x)| and d=|x| are both finite. Then, using fact 3 about powers in finite cyclic groups from Section 2.5, T(e)= e' = T(x)" =T(x") implies r" = e, as T is 1–1. Thus d=|x| divides n=|T(x)|. To go the other way, we can use the fact that T-' is also a group isomorphism by Exercise 3.2.5. So we see in the same way that that n divides d. Why can we do this? It follows that n= d. What happens if x has infinite order?

(3) Suppose n= |T(x)| and d=|x| are both finite. Then, using fact 3 about powers in finite cyclic groups from Section 2.5, T(e)= e' = T(x)" =T(x") implies r" = e, as T is 1–1. Thus d=|x| divides n=|T(x)|. To go the other way, we can use the fact that T-' is also a group isomorphism by Exercise 3.2.5. So we see in the same way that that n divides d. Why can we do this? It follows that n= d. What happens if x has infinite order?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3.2.2

(3) Suppose n=|T(x)| and d=|x| are both finite. Then, using fact 3 about powers in finite cyclic groups from Section 2.5, T(e)= e' = T(x)" =T(x") implies r" = e, as T is 1–1. Thus d=|r| divides n=|T(x)|. To go the other way, we can use the fact that T-1 is also a group isomorphism by Exercise 3.2.5. So we see in the same way that that n divides d. Why can we do this? It follows that n= d. What happens %3D if r has infinite order?
Transcribed Image Text:(3) Suppose n=|T(x)| and d=|x| are both finite. Then, using fact 3 about powers in finite cyclic groups from Section 2.5, T(e)= e' = T(x)" =T(x") implies r" = e, as T is 1–1. Thus d=|r| divides n=|T(x)|. To go the other way, we can use the fact that T-1 is also a group isomorphism by Exercise 3.2.5. So we see in the same way that that n divides d. Why can we do this? It follows that n= d. What happens %3D if r has infinite order?
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