Midterm Review Sheet Algebra Winter
2005
About the Midterm: The midterm will start at 7 pm Wednesday, February 1 in our classroom.
You will have two hours to complete it. It will consist of some definitions to state, some theorems to
state, two theorems to prove, and one or two proof techniques to explain. In addition, there will
be some true/false questions for which you will be expected to briefly justify your answers. There
may be some brief examples to do.
Definitions (You should be able to give an example in each
case)
- Set operations: intersection,
union, difference, Cartesian product
- Complement of a set
- Divides
- Greatest common divisor and Least common multiple
- Relatively prime
- Well ordering principle
- Operation on a set
- Group
- Abelian group
- The group Zn
- Order of a group
- Direct product group
- Subgroup
- Subgroup generated by an element or
a set
- Cyclic subgroup of a group
- Function, domain, range, image
- Preimage of an element under a
function
- Injective function
- Surjective function
- Bijective function
- Inverse function
- Composition of functions
- Permutation group of a set
- The
group Sn
- Dihedral group
Theorems to state
- Division
algorithm
- Euclidean Algorithm
Theorems to prove
- Properties
of divisibility (Theorem 2 in
appendix B)
- Inductive
Principle
- If a
set has an associative operation, then a string of products can be
reassociated in any way.
- In a
group, the identity element and inverses are unique.
- Zn is an abelian group.
- Cancellation
law for groups.
- Theorem
on inverses (Theorem 3, chapter 4)
- There is only one group table for a group of order 1, 2, or 3.
- Subgroup
theorem (a subgroup of a group is itself a group).
- If a nonempty subset S of a finite group G is closed under the operation on G, then it is a subgroup.
Proof techniques to be able to explain
- Proof
by Induction
- Proof
by Strong Induction
- Proof
by Contradiction
- Proof
by Exhaustion (cases)
- Proof of Uniqueness
- Proof using uniqueness