1. Let p E Z be a prime number and set Z, = { e Q : If ged(n, m) = 1, then p {m}.%3De. Let n e Z+ and Un =Zp+2pnShow that any subgroup W of Z(p®) of finite order is of the form Unfor some ne Z+.Hint: choose an element in W whose order is maximal (why is this possible?) and useLagrange's Theorem.

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Answered: 1. Let p € Z be a prime number and set…

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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10.2.11. Let G = D34 = (a,b | a¹7 = b² = e,ba = a¹6b). Let H = (a) and K = = (b). (a) Draw a partial lattice diagram of subgroups of G that includes the subgroups H and K. Include also H¦ K and (H,K). Label the edges with appropriate numbers, and give reasons for what you have done. In what follows, you may refer back to your diagram. (b) Is H
Let G = Z7OU(5). The order of G is O (0, 1) O (1, 0) O (1, 1) O (0, 0) None of the choices
Determine the groups of quadratic residues Qn for all n in the range 10 ≤ n ≤ 20.
1. Let p e Z be a prime number and set Z, = { e Q : If ged(n, m) = 1, then p {m}. %3D d. Show that every proper subgroup of Z(p®) is finite. Hint: Show that if U < Z(p®) is infinite, then U = Z(p®).
2.3. CONJUGACY CLASSES OF S AND An 33 Exercise 55. Write all elements of S3 and S4 and identify the subgroups A and A4 Exercise 56. Prove that An is generated by permutations of type (12k), where 3 ≤ k ≤n. Exercise 57. Prove that the dihedral group De is isomorphic to the Sym- metric group S3. Exercise 58. Prove that Z(Sn) = 1. 2.3 Conjugacy classes of Sn and An By the results of the previous section, every permutation is a product of some cycles. The cyclic structure of a permutation shows the number of the cycles and their orders. For example, if o = (123) (45) (67,8,9), is a permutation in S12, then the cyclic structure of o is 132¹3¹4¹, which means o has 3 cycles of order 1, and on cycle of order2, one cycle of order 3
Let G = U(15) and N = U,(15).f U (n) = {x € U(n) such that x mod k = 1}, then the factor group G/N is: %3D (N, 2N, 4N, 13N) O None of the choices O (N 2N, 4N, 8N ) O(N, 2N, 8N, 1IN} Page 2 of 2 Back Submit NE ersubmit passwords through Google Forms This form was created ins de of Lebaneseintermational University RecoLARUse
19) Let p be a prime and let G be a group of order pn for some integer n ≥ 0. Then G has a subgroup of order pk for all 0 ≤ k≤n.
Let [G:H] be the index of H in G. If 3 points o(G)=9 and o(H)%3D3. Then * [G:H]=1 [G:H]=2 [G:H]=3 [G:H]=4
3. a) In G = (Z54,) find [G:H] for H=. b) Let |G|=8 and G is not cyclic. Show that g 4 = e for every gEG.
3.42. Let G be the set of all sequences of integers (a₁, a2, a3, . . .). 1. Show that G is a group under (a₁, a2, ...) + (b₁, b₂, ...) = (a₁ + b₁, a₂ + b₂,...). 2. Let H be the set of all elements (a₁, a2, ...) of G such that only finitely many a are different from 0 (and (0, 0, 0, ...) E H). Show that H is a subgroup of G.
If m and n are positive integers, and if m is a factor of n, show that Zn contains a subgroup of order m. Does Zn contain more than one subgroup of order m?

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