ABSTRACT ALGEBRAUPVOTE WILL BE GIVEN. PLEASE WRITE THE COMPLETE SOLUTIONS LEGIBLY. NO LONG EXPLANATION NEEDED. (b). Let N = {1,2,3,...,}. Let G = Sym(N) be the group of all bijec-tive functions from N→ N. Then it is clear that G is an infinite group(under composition) as there are infinite bijections from N to N. Con-sider a = (1,2) and b = (2,3). That is a is the function such thata(1) = 2,a(2) = 1 and a (n) = n for all ne N. It is clear a = a-¹since a² = I. Let b = (2, 3) denote the function b(2) = 3, b(3) = 2and b(n) = n for all ne N. Then we havea=¹ba = a oboa= (1, 2) (2, 3) o (1,2)= (1,3)It is clear that a-¹ba = (1,3) b = (2,3).(1)

This is based on simple calcuations of functions. We have used the notations as used for working in…

Answered: (b). Let N = {1,2,3,...,}. Let G =…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 5E

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