(ii) LetK={(), (1 2), (3 4), (5 6), (1 2)(3 4),(1 2) (5 6), (34) (5 6), (1 2)(3 4)(5 6)}with the usual operation of composition of permutations.[You may use without proof the fact that K is a subgroup of the sym-metric group So with |K| = 8.](a) Let L = {(), (1 2)(3 4)}. Compute the right cosets of L in K,and compute the left cosets of L in K.(b) Is there a homomorphism f : K → DĄ with ker ƒ = L? Justifyyour answer.

Answered: Image /qna-images/answer/0513154c-d00e-42d4-b3dd-68f585236441.jpg

Answered: (ii) Let K={(), (1 2), (3 4), (5 6), (1…

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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bb1. Advance maths.
1. Find all Sylow 2-subgroups and Sylow 3-subgroups of the groups De and A4, and verify that they satisfy the second and third Sylow theorem.
1. Which of the following is a nontrivial subgroup of Z3 x Z3 where Z3 is the additive group of integers modulo 3 consisting of {0,1,2} and the Cartesian product is the set of ordered pairs? a) {(0,0),(2,2)} b) {(0,0),(1,2),(2,1)} c) {(0,0),(1,1),(1,2)} d) {0,1} 2. Which of the following is a nontrivial subgroup of the quaternion group of order 8? This group is plus or minus 1, J, K, L where signs multiply as usual, 1 is a multiplicative identity, the squares of J, K, L are -1 and JK=L, KJ=-L, KL=J, LK=-J, LJ=K, JL=-K. The subgroup in question is cyclic. a) {J,-K,L} b) {1,-J,K,L} c) {1,L} d) {1,K,-1,-K} 3. For the following two functions, which of these 4 alternatives is correct? f(x)=x3 -x from the real numbers to itself and g(x)={(1,1),(2,3),(3,1)} from the set {1,2,3} to itself. a) both are one-to-one b) the first, but not the second, is…
(a) Let G be a non-cyclic group of order 121. How many subgroups does G have? Why? (b) Can you generalize your result of the previous part?
2. Let H and K be subgroups of a group G. Show that their intersection HnK is also a subgroup of G. If G is the additive group Z of integers and H and K are the subgroups 67 and 10Z, identify the subgroup 6Z 10Z. What about mZnZ more generally (a proof is not required)?
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