(b) Set n = {0k : kE Z} S C. Since e is a primitive 6th root of unity, ek = e' if and only if k = { mod 6. This gives the second equality in n= {ek : keZ} = (0k: 1sks 6), and gives a natural identification of Q with the set {1, 2, ..., 6}. g11 812) We define an action of G on N as follows. Any g e G is a 2 x 2 matrix g21 g22 For any z en define the action of g on z by: g11z + g12 g.z = g21z + g22 You do not need to show that this gives a well-defined action of G on 2. x = (8 . where e 0 0-1) and Y = () For any keZ find an integer e (not necessarily between 1 and 6) such that X - ek = e . Find and integer m (not necessarily between 1 and 6) such that Y -ek = 0m. (You may find 83 = -1 useful). (ii) With the identification of Q with {1, 2, ..., 6}, write the permutation that X induces on {1, ..., 6} as product of disjoint cycles. Do the same for Y. (iii) Is G transitive on 0? Justify your answer. (iv) Compute the stabiliser of 0° = 1. You may find it useful that every element of G has the form Y"xk for some n, k eZ as well as part (a)(ii).

(b) Set n = {0k : kE Z} S C. Since e is a primitive 6th root of unity, ek = e' if and only if k = { mod 6. This gives the second equality in n= {ek : keZ} = (0k: 1sks 6), and gives a natural identification of Q with the set {1, 2, ..., 6}. g11 812) We define an action of G on N as follows. Any g e G is a 2 x 2 matrix g21 g22 For any z en define the action of g on z by: g11z + g12 g.z = g21z + g22 You do not need to show that this gives a well-defined action of G on 2. x = (8 . where e 0 0-1) and Y = () For any keZ find an integer e (not necessarily between 1 and 6) such that X - ek = e . Find and integer m (not necessarily between 1 and 6) such that Y -ek = 0m. (You may find 83 = -1 useful). (ii) With the identification of Q with {1, 2, ..., 6}, write the permutation that X induces on {1, ..., 6} as product of disjoint cycles. Do the same for Y. (iii) Is G transitive on 0? Justify your answer. (iv) Compute the stabiliser of 0° = 1. You may find it useful that every element of G has the form Y"xk for some n, k eZ as well as part (a)(ii).

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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