All questions are equally weighted. They will be marked for cor-rectness and clarity of explanation.
(a) Let G and H be groups, with g 2 G and h 2 H. Prove that the order
of (g; h) in G H is the least common multiple of o(g) and o(h).
For which pairs of natural numbers m and n is Zm Zn cyclic? Explain.
In S12, let = (5 11) and = (3 4 5 6).
Compute
1 ;
writing the answer as a product of disjoint cycles.
Based on part (a), nd 2 S12 such that
1 = (9 10 11 12):
Let (a1 a2 : : : ak) and (b1 b2 : : : bk) be cycles in Sn. Give a permutation 2 Sn that satis es
1(a1 a2 : : : ak) = (b1 b2 : : : bk);
and explain why your choice of works.
For n 2, let An be the subset of Sn consisting of all even permutations in Sn. Prove that An is a group.
Let n 2. Show that exactly half of the permutations in Sn are even by nding a bijection from the set of all even permutations in Sn to the set of all odd permutations in Sn.
In the group D8, give an algebraic proof that r3j has order 2, and also a geometric proof of the same fact.
For n 4, is Dn cyclic? Explain your answer.
(a) How many generators does the group Z15 have?
Let p and q be distinct primes. How many generators does the group Zpq have?
Is U10 cyclic? Is U12? For each, nd a generator or prove that it is not cyclic.
1
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.
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.
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.
The task of composing nal solutions and writing them up in good copy must be shared equally among all group members (after a collaborative problem-solving process).
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 so-lution.
2
