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2.2.5 Show that S' is not a normal subgroup of S'.

2.2.5 Show that S' is not a normal subgroup of S'.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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of the antipodal point pairs {±q} on the 3-sphere S³. These pairs likewise have a natural product operation, which makes RP' a group-in fact, it is the group SO(3) of rotations of R’. We will show that RP' is not the same group as Sº, because SO(3) is simple and S³ is not. We can see right now that S3 is not simple, by finding a nontrivial normal subgroup. 2.2.4 Show that {±1} is a normal subgroup of S³. However, it turns out that {±1} is the only nontrivial normal subgroup of S'. In particular, the subgroup S' that we found in Section 2.1 is not normal. 2.2.5 Show that S' is not a normal subgroup of S'. 2.3 The groups SU(2) and SO(3)
Transcribed Image Text:of the antipodal point pairs {±q} on the 3-sphere S³. These pairs likewise have a natural product operation, which makes RP' a group-in fact, it is the group SO(3) of rotations of R’. We will show that RP' is not the same group as Sº, because SO(3) is simple and S³ is not. We can see right now that S3 is not simple, by finding a nontrivial normal subgroup. 2.2.4 Show that {±1} is a normal subgroup of S³. However, it turns out that {±1} is the only nontrivial normal subgroup of S'. In particular, the subgroup S' that we found in Section 2.1 is not normal. 2.2.5 Show that S' is not a normal subgroup of S'. 2.3 The groups SU(2) and SO(3)
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