(ii) Let K={(), (1 2), (3 4), (5 6), (1 2)(34), (1 2) (5 6), (3 4) (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? Justify your answer.
(ii) Let K={(), (1 2), (3 4), (5 6), (1 2)(34), (1 2) (5 6), (3 4) (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? Justify your answer.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.8: Some Results On Finite Abelian Groups (optional)
Problem 14E: Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic...
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![(ii) Let
K={(), (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? Justify
your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9deab63c-3625-41fd-8f64-36ab005d6f6f%2F0513154c-d00e-42d4-b3dd-68f585236441%2Fzenxcwr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(ii) Let
K={(), (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? Justify
your answer.
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