Exercise 15.3.2. Fill in the blanks to complete the following proof of Proposition 15.3.1. (a) By the definition of inverse, if g' is an inverse of an element g in a group G, then g. <1> =g. <2>= e. (b) Similarly, if g" is an inverse of g then g· <3> = <4> ·g=e. 514 (e) We may show that g' = g' as follows: g=g. <5> =g' (<6>.g") = (g'.g). <7> = <8> .g" =g" CHAPTER 15 INTRODUCTION TO GROUPS (definition of identity) (part b above, def, of inverse) (associative property of group G) (part a above, def, of inverse) (def. of identity)
Exercise 15.3.2. Fill in the blanks to complete the following proof of Proposition 15.3.1. (a) By the definition of inverse, if g' is an inverse of an element g in a group G, then g. <1> =g. <2>= e. (b) Similarly, if g" is an inverse of g then g· <3> = <4> ·g=e. 514 (e) We may show that g' = g' as follows: g=g. <5> =g' (<6>.g") = (g'.g). <7> = <8> .g" =g" CHAPTER 15 INTRODUCTION TO GROUPS (definition of identity) (part b above, def, of inverse) (associative property of group G) (part a above, def, of inverse) (def. of identity)
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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Please do Exercise 15.3.2 Part ABC. Please show step by step
Transcribed Image Text:Proposition 15.3.1. If g is any element in a group G, then the inverse of
g is unique.
Exercise 15.3.2. Fill in the blanks to complete the following proof of
Proposition 15.3.1.
(a) By the definition of inverse, if g' is an inverse of an element g in a group
G, then g <1> =g² · <2> = e.
(b) Similarly, if g" is an inverse of g then g· <3> = <4> ·g=e.
514
(e) We may show that g' =g" as follows:
g=g. <5>
= g. (<6> ·g")
= (g'.g). <7>
= <8> .g"
=g"
CHAPTER 15 INTRODUCTION TO GROUPS
(definition of identity)
(part b above, def. of inverse)
(associative property of group G)
(part a above, def. of inverse)
(def. of identity)
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