1. In the following exercises, let A be a subset of Mn = {1,2, .., n} and a be a specific élement of A. Determine whether the given set is a subgroup of Sn. (a) {o E Sn | o(a) = a}. (b) {o € Sn | 0(a) E A}. (c) {o E Sn | 0(A) = A} where o(A) = {o(x) | x E A}. (d) {o E S, | o(A) C A} where o(A) = {o(x) | x E A}.

1. In the following exercises, let A be a subset of Mn = {1,2, .., n} and a be a specific élement of A. Determine whether the given set is a subgroup of Sn. (a) {o E Sn | o(a) = a}. (b) {o € Sn | 0(a) E A}. (c) {o E Sn | 0(A) = A} where o(A) = {o(x) | x E A}. (d) {o E S, | o(A) C A} where o(A) = {o(x) | x E A}.

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

(c) Define a permutation of A. (d) Express the product of the cycles (1,2)(4,7,8) (2, 1) (7.2.8.1.5), as a single permutation of the set (1,2,3,4,5,6,7,8).
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
d. Let A = (0,1) and T = {(₁, ¹), (², ¹), G₂²₂), ...}. Show that is open cover for A. Furthermore, determine whether any finite subclass of T is open cover for A.
Let H be the subgroup of S3 generated by the transposition (12). Show that there cannot exists a set X and group action S3 ↷X so that H={σ∈S3 |σx=x∀x∈ X}.
Exercises: (1) Find all subgroupsof (Zs,+s). (2) Let (Zg,+s) be agroup and H=. Is H a subgroup of Zg? (3) If H={0,6,12,18}, show that (H,+24) is a cyclic subgroupof (Z24,+24). Also list the elements of each coset of H in Z24.
Asap
4*. Let f G H be a group homomorphism. Prove: (a) If S G then f(S) 4 f(G) (b) Show by example that S aG need not imply f(S) (c) If T H then f1(T) G. H
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
3.42. 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.
Let f : G → H be a homomorphism with kernel K. Show (a) K is a subgroup of G.(b) For any y ∈ H, f−1(y), the preimage of y, is a left coset of K.
Let H be a subgroup of G, define C(H) the centralizer of H.
i need the answer quickly