### MATH 301 Quiz 1

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.
1. (10 points) Evaluate:  26 mod 11 =

2. (20 points) For the equivalence relation defined by aRb if a mod 11 = b mod 11, what is the equivalence class of 7?

3. 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?

4. (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.

5. (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.