Please help with D-F 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,00+-(: :) ---(: :)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 MEsatisfy 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 seefactor of 8 which is, 1, 2, 4, 8So Total Four subgroups of Q are possible withorder1, 2, 4, 8 respectively.Possible order of subgroups of Q.order of 1Q1 = 8AA =O)A. A- (- 6) (0₁6)(-:-)A². A = (-1 °(-A3, A =So the09) (2.1)-1- E =order of the element A in Q is ųE = ( ! )A². A² = (-E)-(-E)11==( 1²₁1 ) = A ² (From )(!:)E-2(9-%)O

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Answered: The quaternions is the group Q of order…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.1: Definition Of A Group

Problem 51E: Prove that the Cartesian product 24 is an abelian group with respect to the binary operation of...

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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