### 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.

- (30 pts) Draw the subgroup lattice of
**Z**_{30} -- the
integers {0, 1, 2, ... 29} with operation addition mod 30.

- (25 pts) List the elements <(1234)(56)>. In other words, list
the elements of the cyclic group generated by the permutation:

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

- (30 pts) Consider the function Φ from the group
**Z**_{7} 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.