(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

See similar textbooks

Similar questions

3) Let m = i=1 where pi 0 integers. Show that d e N is a divisor of m if and only d = II p* with c; an integer, such i=1 that 0 0 are integers then i=1 ged(m, n) = I[p" min(a;,b;) Pi i=1
From the picture can you prove that is the λσ is one to one and onto? plz with details
Suppose n > 3 is an integer. Prove that in Sn every even permutation is a product of cycles of length 3. Hint: (a, b) (b, c) = (a, b, c) and (a, b)(c, d) = (a, b, c) (b, c, d).
Do all 4 please
29. Let b, k and m be positive integers that satisfy gcd(b, m) = 1 and gcd(k, (m)) = 1, where o(m) is the Euler's Phi function. Prove the uniqueness of the k-th root of b modulo m.
Prove that : a"+ la+1)" 三1 +a(1+()) (mod a (ats) for all a,m,n E Z.
Please show work
Let S be the set [9]{1.2. 9). (a) How many derangements has the set S? (b) How many derangements of S have 9 in the 2nd position and 1 in the last position? (c) List 5 derangements of S having 2 in the 3rd position and 5 in the 2nd position. 101
I don't understand why [0] equals that. Same for [1] and [2]
Please try to solve it in 20 minute