16.17.18.Let R* be the multiplicative group of non zero real numbers and let S =R-{-1} bethe group under the operation defined bya ab=a+b+ ab.Define the mapping f: R* - S by f(r) = r - 1. Show that R* = S.Let f: be defined byf(x) = 3x - 3Prove or disprove: fis an isomorphism of the additive group onto itself.Let G be a group, x E G and H a subgroup of G. Let the subgroup K be defined byK = {k=xhx¹¹ | h eH}.XShow that H = K

As per the guidelines I have solved first problem.

Answered: 16. Let R* be the multiplicative group…

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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16. Let R* be the multiplicative group of non zero real numbers and let S= the group under the operation defined by aab=a+b+ ab. - Define the mapping f: R* S by f(r) = r - 1. Show that R* = S. -{-1} be