Let F be a field. (You may assume F is the field of real numbers) Define f,g : F ∪ {∞} → F ∪ {∞} by f(x) := 1 − x, g(x) := 1/x. Find all the elements of the group generated by f and g under function composition and write out a Cayley table for this group.

Answered: Let F be a field. (You may assume F is…

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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Let F be a field. (You may assume F is the field of real numbers) Define f,g : F ∪ {∞} → F ∪ {∞} by f(x) := 1 − x, g(x) := 1/x. Find all the elements of the group generated by f and g under function composition and write out a Cayley table for this group.

Expert Solution

Step 1

First evaluate the elements of group generated by f and g under function composition by considering the provided functions:

Step 2

Now, evaluate the following:

(fof(x) x))1-(1-x)x
(gog)(x) 8(8(x)
= x
(fog) (x) f(8(x))1-
x
1
(sof)(x)8((x))
1-x

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