MATH 301 Quiz 2

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. 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.





  2. (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.






  3. (20 pts) Recall that R180 is in Z(D4). Since D4 = {R0, R90, R180, R270, H, V, D, D'} is a group, it is closed under group operation. Which group element equals the combination H R180 H-1?







  4. (20 points) Suppose G is a group and a is an element of G. If |a| = 8, does <a4> = <a6>? Justify your answer. (Hint: you can either compare the elements of the two cyclic groups or use a theorem from Chapter 4.)








Bonus (5 pts) True or false: V is an element of the centralizer C(V) in D4. Justify your answer.