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.
(10 points) Evaluate: 26 mod 11 =
(20 points) For the equivalence relation defined by aRb if a mod 11 = b mod
11, what is the equivalence class of 7?
The set {1, 2, 3, 4, 5, 6} together with the
operation "multiplication mod 7" forms a group G.
a) (10 pts) What is the
identity element in this group?
b) (10 pts) What is the inverse of the element 5 in this group?
c) (10 pts) What is the order of the element 4 of this group?
(20 points) List the properties that a set G together with an operation *
must have in order to be a group, or else give the definition of a group.
(20 points) Consider the set {1, 2, 3, 4, 5, 6, 7} and the operation
"multiplication mod 8". Prove or disprove that these combine to make a
group.
Bonus (5 points) Suppose a and b are group elements and e is the
group identity. Prove that if ba = e then it must be true that ab = e.