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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
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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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: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
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
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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