### MATH 301 Quiz 3

Name:
You may use a calculator on this quiz. You may not use a cell phone or computer. To receive full credit please show your work carefully and give justifications for your answers. If you find that you are spending a lot of time on one problem, leave it blank and move on to the next. There are questions on both sides of this page.
1. (30 pts) Draw the subgroup lattice of Z30 -- the integers {0, 1, 2, ... 29} with operation addition mod 30.

2. (25 pts) List the elements <(1234)(56)>. In other words, list the elements of the cyclic group generated by the permutation:
 [ 1 2 3 4 5 6 ] [ 2 3 4 1 6 5 ].

3. (15 pts) Find a permutation of degree six in the symmetric group S5 -- the group of permutations of the set {1, 2, 3, 4, 5}.

4. (30 pts) Consider the function Φ from the group Z7 to itself defined by Φ(k) = (7 - k) mod 7. Prove that Φ is an isomorphism; in other words, show that Φ is an automorphism of the group {0, 1, 2, ... 6} with operation addition mod 7.

Bonus (5 pts) For any group G and any element g of G, define a map γ : G -> G such that γ(a) = ga for all elements a of G. True or false: &gamma is an isomorphism of G. Explain how you arrived at your answer.