Let G = {±1, ±i, ±j, ±k}, where i? = j² = k² = –1, -i = (-1)i, 12 = (-1)2 = 1, ij = - ji = k, jk = –kj = i, and ki = –ik = j. a. Show that H = {1, – 1} 4G. b. Construct the Cayley table for G/H. Is GIH isomorphic to Z4 or ZOZ? (The rules involving i, j, and k can be remembered by using the cir- cle below. Going clockwise, the product of two consecutive elements is the third one. The same is true for going counterclockwise, except that we ob- tain the negative of the third element. This group is called the quater- nions. It was invented by William Hamilton in 1843. The quaternions are used to describe rotations in three-dimensional space, and they are used in physics. The quaternions can be used to extend the complex numbers in a natural way).
Let G = {±1, ±i, ±j, ±k}, where i? = j² = k² = –1, -i = (-1)i, 12 = (-1)2 = 1, ij = - ji = k, jk = –kj = i, and ki = –ik = j. a. Show that H = {1, – 1} 4G. b. Construct the Cayley table for G/H. Is GIH isomorphic to Z4 or ZOZ? (The rules involving i, j, and k can be remembered by using the cir- cle below. Going clockwise, the product of two consecutive elements is the third one. The same is true for going counterclockwise, except that we ob- tain the negative of the third element. This group is called the quater- nions. It was invented by William Hamilton in 1843. The quaternions are used to describe rotations in three-dimensional space, and they are used in physics. The quaternions can be used to extend the complex numbers in a natural way).
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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Question
Transcribed Image Text:Let G = {±1, ±i, ±j, ±k}, where i? = j² = k² = –1, -i = (-1)i,
12 = (-1)2 = 1, ij = - ji = k, jk = –kj = i, and ki = –ik = j.
a. Show that H = {1, – 1} 4G.
b. Construct the Cayley table for G/H. Is GIH isomorphic to Z4 or
ZOZ?
(The rules involving i, j, and k can be remembered by using the cir-
cle below.
Transcribed Image Text:Going clockwise, the product of two consecutive elements is the third
one. The same is true for going counterclockwise, except that we ob-
tain the negative of the third element. This group is called the quater-
nions. It was invented by William Hamilton in 1843. The quaternions
are used to describe rotations in three-dimensional space, and they are
used in physics. The quaternions can be used to extend the complex
numbers in a natural way).
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