Let and , the subgroup of fourth roots of unity. (a) Characterize all left cosets of...

Question

Question

Let G =C and H={+1, i) , the subgroup of fourth roots of unity.
(a) Characterize all left cosets of H.
(b) Prove or disprove that G/H isomorphic to G.

math Advanced-Math

Add a comment Improve this question Transcribed image text

Answer 1

Answer #1

O All cosets all of the foom. ZH={+2, liz] hele zec*. thus for 2=arib, a, b e R. we have that ZH= Carib, -a-ib, ia-b, b-ia] (

Add a comment

Answer 2

Similar Homework Help Questions

Let G be a finite group and let H be a subgroup of G. Show using double cosets that there is a su...

Let G be a finite group and let H be a subgroup of G. Show using double cosets that there is a subset T of G which is simultaneously a left transversal for H and a right transversal for H.
(5) Let G be a group, and let H be a subgroup of G. Define a...

(5) Let G be a group, and let H be a subgroup of G. Define a relation ~ on G as follows: X~ · y if x-ly E H. Prove that this is an equivalence relation, and that the equivalence classes of the relation are the left cosets of H.
Q. 5 5. Let H G be a subgroup and suppose that H,g2H.....gH are the distinet...

Q. 5 5. Let H G be a subgroup and suppose that H,g2H.....gH are the distinet left cosets of H in G. Prove that gH - Hg for all g e G if and only if g.H Hg,, for all 2 sisr

(2) (a) Prove that the set G = {+1, £i} is a finite subgroup of the...

(2) (a) Prove that the set G = {+1, £i} is a finite subgroup of the multi- plicative group CX of nonzero complex numbers, and that the set H = {E1} is a finite subgroup of {+1, £i}. (b) Compute the index of H in G. (c) Compute the set of left cosets G/H.
4. Let H be a subgroup of a group G and let a, b e H....

4. Let H be a subgroup of a group G and let a, b e H. Using the definition of cosets, prove that Ha= Hb if and only if ab-EH.
List all left cosets of the subgroup of and find the index . We were unable...

List all left cosets of the subgroup of and find the index . We were unable to transcribe this imageWe were unable to transcribe this image(Z: 3Z)

I help help with 34-40 33. I H is a subgroup of G and g G, prove that gHg-1 is a subgroup of G. Also, prove that the intersection of gH for all g is a normal subgroup of G. 34. Prove that 123)(m...

I help help with 34-40 33. I H is a subgroup of G and g G, prove that gHg-1 is a subgroup of G. Also, prove that the intersection of gH for all g is a normal subgroup of G. 34. Prove that 123)(min-1n-)1) 35. Prove that (12) and (123 m) generate S 36. Prove Cayley's theorem, which is the followving: Any finite group is isomorphic to a subgroup of some S 37. Let Dn be the dihedral group of...
(8) Let G be a group and let H be a subgroup of G. Prove that...

(8) Let G be a group and let H be a subgroup of G. Prove that the right cosets of H partition G, that is, G= U Hy HYEH\G and, if y, y' E G and Hyn Hy' + 0, then Hy= Hy'.
18. Let N be a normal subgroup of a finite group G, and let Nxi, ....

18. Let N be a normal subgroup of a finite group G, and let Nxi, . N be a for complete list of (disjoint) right cosets. Prove that, as sets, Nx, Nz all i and j Nz,

Let H denote the smallest subgroup of S(5) containing the cycles (1 2 3) and (3,5)....

Let H denote the smallest subgroup of S(5) containing the cycles (1 2 3) and (3,5). (a) List all the elements in H. (b) How many left cosets of H in G are there?

Let and , the subgroup of fourth roots of unity. (a) Characterize all left cosets of...

Homework Answers

Add Answer to:
Let and , the subgroup of fourth roots of unity. (a) Characterize all left cosets of...

Post as a guest

Earn Coins

Let G be a finite group and let H be a subgroup of G. Show using double cosets that there is a su...

(5) Let G be a group, and let H be a subgroup of G. Define a...

Q. 5 5. Let H G be a subgroup and suppose that H,g2H.....gH are the distinet...

(2) (a) Prove that the set G = {+1, £i} is a finite subgroup of the...

4. Let H be a subgroup of a group G and let a, b e H....

List all left cosets of the subgroup of and find the index . We were unable...

I help help with 34-40 33. I H is a subgroup of G and g G, prove that gHg-1 is a subgroup of G. Also, prove that the intersection of gH for all g is a normal subgroup of G. 34. Prove that 123)(m...

(8) Let G be a group and let H be a subgroup of G. Prove that...

18. Let N be a normal subgroup of a finite group G, and let Nxi, ....

Let H denote the smallest subgroup of S(5) containing the cycles (1 2 3) and (3,5)....

Let and , the subgroup of fourth roots of unity. (a) Characterize all left cosets of...

Homework Answers

Add Answer to: Let and , the subgroup of fourth roots of unity. (a) Characterize all left cosets of...

Post as a guest

Earn Coins

Add Answer to:
Let and , the subgroup of fourth roots of unity. (a) Characterize all left cosets of...