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2. In this exercise, we will construct a new finite group of order 8 using the quater- nions. The quaternions are a number system that extends the complex numbers. Read the following excellent articles to learn about them before moving on: http://shann.idv.tw/Lite/essay/9908.pdf http://shann.idv.tw/Lite/essay/9909.pdf The quaternion group is the group Q which consists of the eight quaternions {±1, ±i, ±j, ±k}, using the quaternion multiplication as the group operation. (a) Show that Q D4. (b) Find all subgroups of Q, and show that they are all normal in Q.

2. In this exercise, we will construct a new finite group of order 8 using the quater- nions. The quaternions are a number system that extends the complex numbers. Read the following excellent articles to learn about them before moving on: http://shann.idv.tw/Lite/essay/9908.pdf http://shann.idv.tw/Lite/essay/9909.pdf The quaternion group is the group Q which consists of the eight quaternions {±1, ±i, ±j, ±k}, using the quaternion multiplication as the group operation. (a) Show that Q D4. (b) Find all subgroups of Q, and show that they are all normal in Q.

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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2. In this exercise, we will construct a new finite group of order 8 using the quater- nions. The quaternions are a number system that extends the complex numbers. Read the following excellent articles to learn about them before moving on: http://shann.idv.tw/Lite/essay/9908.pdf http://shann.idv.tw/Lite/essay/9909.pdf The quaternion group is the group which consists of the eight quaternions {±1, ±i, ±j, ±k}, using the quaternion multiplication as the group operation. (a) Show that Q ‡ D4. (b) Find all subgroups of Q, and show that they are all normal in Q.
Transcribed Image Text:2. In this exercise, we will construct a new finite group of order 8 using the quater- nions. The quaternions are a number system that extends the complex numbers. Read the following excellent articles to learn about them before moving on: http://shann.idv.tw/Lite/essay/9908.pdf http://shann.idv.tw/Lite/essay/9909.pdf The quaternion group is the group which consists of the eight quaternions {±1, ±i, ±j, ±k}, using the quaternion multiplication as the group operation. (a) Show that Q ‡ D4. (b) Find all subgroups of Q, and show that they are all normal in Q.
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