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.
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 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?
(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.