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The Quaternions The group of quaternions is Q = {1, -1, i, -i, j, -j, k, -k} where multiplication is given in the Cayley table below. | | 1 | -1 | i | -i | j | -j | k | -k | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | 1 | 1 | -1 | i | -i | j | -j | k | -k | | -1 | -1 | 1 | -i | i | -j | j | -k | k | | i | i | -i | -1 | 1 | k | -k | -j | j | | -i | -i | i | 1 | -1 | -k | k | j | -j | | j | j | -j | -k | k | -1 | 1 | i | -i | | -j | -j | j | k | -k | 1 | -1 | -i | i | | k | k | -k | j | -j | -i | i | -1 | 1 | | -k | -k | k | -j | j | i | -i | 1 | -1 | 1. List all of the cyclic subgroups (including their elements) of Q. 2. Make a subgroup lattice for Q. (Hint: all of the proper subgroups are cyclic.) 

          The Quaternions

The group of quaternions is Q = {1, -1, i, -i, j, -j, k, -k} where multiplication is given in the Cayley table below.

| | 1 | -1 | i | -i | j | -j | k | -k |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| 1 | 1 | -1 | i | -i | j | -j | k | -k |
| -1 | -1 | 1 | -i | i | -j | j | -k | k |
| i | i | -i | -1 | 1 | k | -k | -j | j |
| -i | -i | i | 1 | -1 | -k | k | j | -j |
| j | j | -j | -k | k | -1 | 1 | i | -i |
| -j | -j | j | k | -k | 1 | -1 | -i | i |
| k | k | -k | j | -j | -i | i | -1 | 1 |
| -k | -k | k | -j | j | i | -i | 1 | -1 |

1. List all of the cyclic subgroups (including their elements) of Q.
2. Make a subgroup lattice for Q. (Hint: all of the proper subgroups are cyclic.)
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The Quaternions The group of quaternions is Q = 1, -1, i, -i, j, -j, k, -k where multiplication is given in the Cayley table below. | | 1 | -1 | i | -i | j | -j | k | -k | | :— | :— | :— | :— | :— | :— | :— | :— | :— | | 1 | 1 | -1 | i | -i | j | -j | k | -k | | -1 | -1 | 1 | -i | i | -j | j | -k | k | | i | i | -i | -1 | 1 | k | -k | -j | j | | -i | -i | i | 1 | -1 | -k | k | j | -j | | j | j | -j | -k | k | -1 | 1 | i | -i | | -j | -j | j | k | -k | 1 | -1 | -i | i | | k | k | -k | j | -j | -i | i | -1 | 1 | | -k | -k | k | -j | j | i | -i | 1 | -1 | 1. List all of the cyclic subgroups (including their elements) of Q. 2. Make a subgroup lattice for Q. (Hint: all of the proper subgroups are cyclic.)

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Mathematical Methods for Physics and Engineering
Mathematical Methods for Physics and Engineering
K. F. Riley, M. P. Hobson, S. J. Bence 3rd Edition
Chapter 28
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The group of quaternions is Q = {1, -1, i, -i, j, -j, k, -k} where multiplication is given in the Cayley table below: 1 | -1 | i | -i | j | -j | k | -k --- | --- | --- | --- | --- | --- | --- | --- 1 | 1 | -1 | i | -i | j | -j | k | -k -1 | -1 | 1 | -i | i | -j | j | -k | k i | i | -i | -1 | 1 | k | -k | -j | j -i | -i | i | 1 | -1 | -k | k | j | -j j | j | -j | -k | k | -1 | 1 | i | -i -j | -j | j | k | -k | 1 | -1 | -i | i k | k | -k | j | -j | -i | i | -1 | 1 -k | -k | k | -j | j | i | -i | 1 | -1 1. List all of the cyclic subgroups (including their elements) of Q. 2. Make a subgroup lattice for Q. (Hint: all of the proper subgroups are cyclic.)
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6. * The quaternion group Q of order 8 has elements 1, -1, i, -i, j, -j, k, -k, with the following multiplication table. 1 -1 i -i j -j k -k 1 1 -1 i -i j -j k -k -1 -1 1 -i i -j j -k k i i -i -1 1 k -k -j j -i -i i 1 -1 -k k j -j j j -j -k k -1 1 i -i -j -j j k -k 1 -1 -i i k k -k j -j -i i -1 1 -k -k k -j j i -i 1 -1 (a) What is the centre of Q? (b) What is the centralizer in Q of the element -1? (c) What is the centralizer in Q of the element i? Remark: Multiplication in Q satisfies: i² = j² = k² = -1 = ijk. The quaternion group lives inside the quaternion division algebra which was famously discovered by William Rowan Hamilton in Dublin in 1843. It extends the set of complex numbers, and has three independent square roots of -1 (i, j and k) instead of just i. The multiplication of quaternions is not commutative.

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Transcript

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00:01 Hello everyone, so here h is equal to minus 413, minus 1 .7, 05, 5 by 2, 0, 3 minus 5.
00:12 F is equal to minus 4 minus 7, minus 2 minus 5, 0 minus 3, 30 5296.
00:31 So the domain is minus 4 minus 1 0 5 by 2 3.
00:39 Therefore this is the h domain and this is the f domain...
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