Math 212, Assignment 2

All questions are equally weighted. They will be marked for correctness and clarity of explanation.

1. (a) Let G and H be groups, with g ∈ G and h ∈ H. Prove that the order

of (g, h) in G ⊕ H is the least common multiple of o(g) and o(h).

(b) For which pairs of natural numbers m and n is Zm⊕Zn cyclic? Explain.

2. In S12, let σ = (5 11) and π = (3 4 5 6).

(a) Compute

σ

−1πσ,

writing the answer as a product of disjoint cycles.

(b) Based on part (a), find τ ∈ S12 such that

τ

−1πτ = (9 10 11 12).

(c) Let (a1 a2 . . . ak) and (b1 b2 . . . bk) be cycles in Sn. Give a permutation

τ ∈ Sn that satisfies

τ

−1

(a1 a2 . . . ak)τ = (b1 b2 . . . bk),

and explain why your choice of τ works.

3. For n ≥ 2, let An be the subset of Sn consisting of all even permutations in

Sn. Prove that An is a group.

4. Let n ≥ 2. Show that exactly half of the permutations in Sn are even by

finding a bijection from the set of all even permutations in Sn to the set of

all odd permutations in Sn.

5. In the group D8, give an algebraic proof that r

3

j has order 2, and also a

geometric proof of the same fact.

6. For n ≥ 4, is Dn cyclic? Explain your answer.

7. (a) How many generators does the group Z15 have?

(b) Let p and q be distinct primes. How many generators does the group

Zpq have?

8. Is U10 cyclic? Is U12? For each, find a generator or prove that it is not

cyclic.

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Rules for group assignments. Make sure you follow the universal rules for

group assignments (below) and any additional rules/procedures laid out in your

Group Contract.

1. Each group member is expected to contribute to the best of their ability, and

assignment submissions should only include the names of group members

who meet this expectation.

2. Each group member should be able to explain the group’s solution to me

and answer any questions I may have about it. It is the whole group’s

responsibility to ensure that this standard is met.

3. The task of composing final solutions and writing them up in good copy

must be shared equally among all group members (after a collaborative

problem-solving process).

4. After good copy solutions are complete, they should be shared among all

group members to be double-checked and proofread. This should be done

in advance of the due date, to allow time for any necessary corrections.

Corrections should be completed by the person who wrote the original solution.

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