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.

- Let G be the group {1, 2, 3, 4, 5, 6} of integers with group
operation
**multiplication**mod 7.a) (10 pts) What is |2|?

b) (20 pts) Is this group cyclic? Justify your answer. (Hint: find <x> for different elements x of G.)

c) (10 pts) Is the group G Abelian? Justify your answer.

- (20 pts) Is the set {0, 2, 3, 6} a subgroup of the group {0, 1,
2, 3, 4, 5, 6, 7) of integers with operation addition mod 8? Justify
your answer.
- (20 pts) Recall that R
_{180}is in Z(D_{4}). Since D_{4}= {R_{0}, R_{90}, R_{180}, R_{270}, H, V, D, D'} is a group, it is closed under group operation. Which group element equals the combination H R_{180}H^{-1}? - (20 points) Suppose G is a group and a is an element of G. If
|a| = 8, does <a
^{4}> = <a^{6}>? Justify your answer. (Hint: you can either compare the elements of the two cyclic groups or use a theorem from Chapter 4.)