MATH 128A San Jose State University Does There Exist an Automorphism Questions

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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.