Recall that GL(2, C) is the group of invertible 2x 2 matrices over C (i.e. matrices with non-vanishing determinant) equipped with multiplication of matrices as the group operation and I = 6 )a the identity element. Consider the following matrices in GLI2, C): x= : ) and Y = G where 8 = e Notice that e is a primitive 6th root of unity, so 86 =1 and ek + 1, for 1sk<6. Let K denote the (cyclic) subgroup of GL(2, C) generated by X and let H denote the (cyclic) subgroup generated by Y, namely K= and H = . Set G= HK = {X: y €H, XE K} = {Y*x* : k, n EZ}C GL2, C).

Recall that GL(2, C) is the group of invertible 2x 2 matrices over C (i.e. matrices with non-vanishing determinant) equipped with multiplication of matrices as the group operation and I = 6 )a the identity element. Consider the following matrices in GLI2, C): x= : ) and Y = G where 8 = e Notice that e is a primitive 6th root of unity, so 86 =1 and ek + 1, for 1sk<6. Let K denote the (cyclic) subgroup of GL(2, C) generated by X and let H denote the (cyclic) subgroup generated by Y, namely K= and H = . Set G= HK = {X: y €H, XE K} = {Yx : k, n EZ}C GL2, C).

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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