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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.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.7: Distinguishable Permutations And Combinations
Problem 16E
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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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