The quaternions is the group Q of order 8 consisting of the matrices in GL2(C) Q = {E, A, A², A³, B, BA, BA2, BA³} where E is the identity matrix
The quaternions is the group Q of order 8 consisting of the matrices in GL2(C) Q = {E, A, A², A³, B, BA, BA2, BA³} where E is the identity matrix
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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Please help with D-F
Transcribed Image Text:1. The quaternions is the group Q of order 8 consisting of the matrices in GL2(C)
Q = {E, A, A², A³, B, BA, BA2, BA³}
where E is the identity matrix,
0
0
+-(: :) ---(: :)
A =
and B =
-1
(a) What are the possible orders of subgroups of Q?
(b) What is the order of the element A E Q?
(c) Prove that -E is the only element in Q of order 2, and that all other elements ME
satisfy M² = E
(d) Which subgroups of Q are normal?
(e) Show that (A) < Q?
(f) Find the left cosets of (A) in Q
Transcribed Image Text:Sol" Q = {E, A, A², A³, B, BA, BA², BA³S
Ⓒ
(6)
Ⓒ
We see
factor of 8 which is, 1, 2, 4, 8
So Total Four subgroups of Q are possible with
order
1, 2, 4, 8 respectively.
Possible order of subgroups of Q.
order of 1Q1 = 8
A
A =
O
)
A. A- (- 6) (0₁6)
(-:-)
A². A = (-1 °
(-
A3, A =
So the
0
9) (2.1)
-1
- E =
order of the element A in Q is ų
E = ( ! )
A². A² = (-E)-(-E)
11
=
=
( 1²₁1 ) = A ² (From )
(!:)
E
-
2
(9-%)
O
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