Let G (Z,+) and let n be a positive integer. Let H nZ = {na : a E Z}. We know that H is a subgroup of G. Describe the right cosets of H and G and find |G H.

Q: 1. Which of the following subgroups of Z27 has order 3? В. (9) А. (3) С. (6) D. (12) 2. Which of the…

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Q: Let C2 = {±1} be a subgroup of R". Prove that R = R+ x C2 where R is the set of real numbers.

A: Please check step 2 and 3 for the solution.!

Q: Let G = 64. (a.) Let a = (123) (14). Write a ¹2 in disjoint cycle form. (b.) Consider the subgroup H…

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Q: Find all homomorphic images of dihedral group, D3.

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Q: Let M and N be normal subgroups of G. Show that MN is also a normal subgroup of G

A: It is given that M and N are normal subgroups of G. implies that,

Q: Let G = A4. (3) Find the subgroup H of order 4. (4) List all left cosets of H in G.

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Q: Let G=U(16),H={1,15},andK={1,9} . Are H and K isomorphic?Are G/H and G/K isomorphic?

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Q: ow many subgroups of order 3 does D3 have? Write a number of subgroups only.

A: D3={1 , r , r2 , f , rf , r2f }  Or=3 Of=2    rmf=frn-m   n=3 , 0≤m≤3 O(rif)=2   0≤i≤2 3-order…

Q: 3. Let T E S3 denote the 3-cycle (123). Show that the subgroup (7) is normal in S3.

A: We know that any subgroup of a group which has index 2 is called normal subgroup of the group. Here…

Q: Suppose that a subgroup H of S5 contains a 5-cycle and a 2-cycle.Show that H = S5.

A: It is given that, H be a subgroup of S5 which contains a 5-cycle and a 2-cycle. We have to show…

Q: Let f: R→ R be given by f(x) = 2x. Let g: Z → Z be given by g(x) = 2x. Select the true statements.…

A: Given: f:R→R is given by fx=2x.g:Z→Z is given by gx=2x. To Do: We have to select the true statements…