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Math 128a, problem set 06

Outline due: Wed Oct 07

Due: Mon Oct 12

Last revision due: Mon Nov 23

Problems to be done, but not turned in: (Ch. 6) 7–67 odd; (Ch. 7) 1–17 odd.

Fun: (Ch. 6) 64, 66.

Problems to be turned in:

1. Does there exist an automorphism ϕ : Z70 → Z70 such that ϕ(17) = 21? If so, describe

all such ϕ as precisely as possible, with proof; if not, prove that no such ϕ exists.

2. Consider the groups U (15), U (20), and U (24). For any two of them that you think

are not isomorphic, prove that they are not isomorphic.

3. Find three groups G, H, K of order 24 such that G 6≈ H, H 6≈ K, and G 6≈ K. Prove

your result.

4. Consider the group D6 , using our standard notation.

(a) Let K = {e, F12 } = hF12 i. List all of the left cosets of K and all of the right

cosets of K.

(b) Let H = {e, R120 , R240 } = hR120 i. List all of the left cosets of H and all of the

right cosets of H. Do you see any significant qualitative differences between this

example and the previous one? Explain.

5. Let H = 5Z = {n ∈ Z | n = 5k for some k ∈ Z}. List all of the cosets of H in Z. How

many are there? Generalize as much as possible, both in terms of numbers of cosets

and what those cosets are.

6. Let G be a group, and let H and K be subgroups of G such that |H| = 60 and

|K| = 70. What are the possibilities for the order of H ∩ K? Generalize.

7. (a) Let G be a group such that every nontrivial element of G has order 2. Prove

that G is abelian.

(b) Now let G be a group of order 8. Prove that if G is not abelian, then G must

have an element of order 4.

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