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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} = {Y*x* : k, n EZ}C GL2, C).

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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Recall that GL(2, C ) is the group of invertible 2 x 2 matrices over C (i.e. matrices with non-vanishing determinant) equipped with multiplication of matrices as the group operation and I =| 6 )»= as the identity element. Consider the following matrices in GL(2, C): x= (8 ) and Y = (G where e = es 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 = <X> and H = <Y>. Set G= HK = (VX: y EH, XE K} = {Y*x" : k, nEZ}C GL(2, C).
Transcribed Image Text:Recall that GL(2, C ) is the group of invertible 2 x 2 matrices over C (i.e. matrices with non-vanishing determinant) equipped with multiplication of matrices as the group operation and I =| 6 )»= as the identity element. Consider the following matrices in GL(2, C): x= (8 ) and Y = (G where e = es 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 = <X> and H = <Y>. Set G= HK = (VX: y EH, XE K} = {Y*x" : k, nEZ}C GL(2, C).
(iv) Deduce that HKHCK for every h E H, hence H C NGL2.0(K) (the normaliser of K). (v) Recall that G = HK. Use the previous part of the question, or otherwise, to explain why G is a subgroup of GL(2, C) and to prove that Ke G. (vi) Which matrices does Hn K contain? What is |Hn K|? (vii) What is the order of G? (viii) Find a Sylow 3-subgroup of G. (ix) Find a Sylow 2-subgroup of G. Is it normal in G? How many Sylow 2-subgroups does G have?
Transcribed Image Text:(iv) Deduce that HKHCK for every h E H, hence H C NGL2.0(K) (the normaliser of K). (v) Recall that G = HK. Use the previous part of the question, or otherwise, to explain why G is a subgroup of GL(2, C) and to prove that Ke G. (vi) Which matrices does Hn K contain? What is |Hn K|? (vii) What is the order of G? (viii) Find a Sylow 3-subgroup of G. (ix) Find a Sylow 2-subgroup of G. Is it normal in G? How many Sylow 2-subgroups does G have?
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