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Exercise 15.3.6. Fill in the blanks to complete the proof of Proposi- tion 15.3.4

Exercise 15.3.6. Fill in the blanks to complete the proof of Proposi- tion 15.3.4

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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Please do Exercise 15.3.6

the definition of inverse (from Proposition 15.3.1, we know the inverse is unique). First, we have: (ab)(b¹a-¹)= a(bb-¹)a-¹ (associative property of group G) = aca-¹ (def. of inverse) -1 =aa¹ (def. of identity) = e. (def. of inverse) The remainder of the proof is left as an exercise: Exercise 15.3.6. Fill in the blanks to complete the proof of Proposi- tion 15.3.4 (b¹a¹)(ab)=b¹(a¹a)b =b-¹eb =b-¹b = e. By repeated application of Proposition 15.3.4, we may find the inverse of the product of multiple group elements, for example: (abcd)¹=d²¹c²¹b²¹a²¹. Proposition 15.3.4 shows that in general, when finding inverses of prod- ucts it is necessary to take the products of inverses in reverse order. One might ask, Is it ever the case that it's not necessary to reverse the order? Glad you asked! We address this question in the following exercise:
Transcribed Image Text:the definition of inverse (from Proposition 15.3.1, we know the inverse is unique). First, we have: (ab)(b¹a-¹)= a(bb-¹)a-¹ (associative property of group G) = aca-¹ (def. of inverse) -1 =aa¹ (def. of identity) = e. (def. of inverse) The remainder of the proof is left as an exercise: Exercise 15.3.6. Fill in the blanks to complete the proof of Proposi- tion 15.3.4 (b¹a¹)(ab)=b¹(a¹a)b =b-¹eb =b-¹b = e. By repeated application of Proposition 15.3.4, we may find the inverse of the product of multiple group elements, for example: (abcd)¹=d²¹c²¹b²¹a²¹. Proposition 15.3.4 shows that in general, when finding inverses of prod- ucts it is necessary to take the products of inverses in reverse order. One might ask, Is it ever the case that it's not necessary to reverse the order? Glad you asked! We address this question in the following exercise:
An important property of inverses is: Proposition 15.3.4. Let G be a group. If a, b = G, then (ab)-¹ = b¯¹a-¹. Remark 15.3.5. We've actually seen this property before, in the permu- tations chapter: recall that for two permutations and 7, we showed that (OT)-¹ =7-¹0-¹ A PROOF. By the inverse property, 3a-1,b-1 € G. By the closure property, ab € G and b¹a-¹ G. So we only need to verify that b¹a-¹ satisfies
Transcribed Image Text:An important property of inverses is: Proposition 15.3.4. Let G be a group. If a, b = G, then (ab)-¹ = b¯¹a-¹. Remark 15.3.5. We've actually seen this property before, in the permu- tations chapter: recall that for two permutations and 7, we showed that (OT)-¹ =7-¹0-¹ A PROOF. By the inverse property, 3a-1,b-1 € G. By the closure property, ab € G and b¹a-¹ G. So we only need to verify that b¹a-¹ satisfies
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