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State the Well Ordering Principle
Every nonempty set of positive integers contains a smallest element.

State the Division Algorithm
Let a and b be integers where b > 0. Then there exists unique integers q&r such that: a = bq + r where 0< r < b

Linear Combination
For any nonzero integers a and b, there exists integers s & t such that gcd(a,b) = as + bt. Moreover, gcd(a,b) is the smallest positive integer of the form as + bt.

If a and b are relatively prime...
... then there exists integers s and t such that as + bt = 1.

Define Euclid's Lemma
If p is a prime that divides ab, then p divides a or p divides b.

Define the GCD
 Greatest Common Divisor
 The GCD of two nonzero integers a and b is the largest of all common divisors of a and b. We denote this integer gcd(a,b).

Define the LCM
 Least Common Multiple
 The LCM of two nonzero integers a and b is the smallest positive integer that is a multiple of both a and b. We denote this as lcm(a,b).

Define an Equivalence Relation
 An equivalence relation on a set S is a set of ordered pairs R of S such that
 ~(a,a) is an element of R for all a element of S (reflexive)
 ~(a,b) is an element of R then (b,a) is an element of R (symmetric)
 ~If (a,b) is an element of R and (b,c) is an element of R then (a,c) is an element of R (Transitive)

Partition of a Set
A partition of a set S is a collection of nonempty disjoint subsets of S whose union is S.

Define a function
A function (mapping) is a rule that assigns each a of A a unique element b of B.

Dihedral Group
a gropu denoted D_{n} of order 2n

What is a group?
Let G be a set under a binary operation. A group assigns to each ordered pair (a,b) of elements of G an element in G denoted by ab.

What is a binary operation?
A binary operation on set G is a function that assigns each ordered pair of elements of G an element of G.

What three properties must a group satisfy?
 1. Associativity. i.e. (ab)c = a(bc) for all a,b,c in G.
 2. Identity. i.e. there is an element called e such that ae = ea = a for all a in G.
 3. Inverses. i.e. there is an element b such that ab=ba=e.

What does it mean to say a group is Abelian?
The group has the property that ab = ba for every pair of elements a and b.

What is the Uniqueness of the Identity theorem?
In a group G, there is only one identity element.

State the Cancellation Theorem
Right and Left cancellation holds such that ba=ca implies b=c, and ab=ac implies b=c.

State the Uniqueness of Inverses Theorem
For each element a in a group G there is a unique element b in G such that ab=ba=e.

What is the SocksShoesProperty?
 For group elements a and b,
 (ab)^{1} =b^{1}a^{1}

Define the Order of a Group
The order of a group (finite or infinite) is the number of elements in this group. We denote the order of group G as G.

Define the Order of an Element
Let G be a group and g be an element of G. The order of an element g in G is the smallest possible integer n such that g^{n}=e. If no such n exists then we say g has infinite order. Denoted by g.

Define a Subgroup
If a subset H of a group G is also a group under the same operation of G, we say that H is a subgroup of G. Denoted H<G

What are the steps in the OneStep Subgroup test?
 1. Identify the property P that distinguishes the elements of H
 2. Prove that the identity has the property P
 3. Assume that two elements a and b have property P
 4. Use the assumption that a and b have property P to show that ab^{1 }also has property P

State the OneStep Subgroup test
 Let G be a group and H a nonempty subset of G.
 (multiplication) If ab^{1} is in H whenever a and b are in H, then H is a subgroup of G.
 (additive) If a  b is in H whenever a and b are in H, then H is a subgroup of G.

State the TwoStep Subgroup Test
 Let G be a group and let H be a nonempty subset of G.
 If ab is in H whenever a and b are in H, and a^{1} is in H whenever a is in H, then H is a subgroup of G.

State the FiniteSubgroup Test
When is this used?
Let H be a nonempty finite subset of a group G. If H is closed under the operation of G, then H is a subgroup of G.
used when we have finite groups

What is <a> ?
 Cyclic Group  Let G be a group, and let a be any element of g. Then, <a> is a subgroup denoted by
 <a> = {a^{n}  n is an element of Z}

How do we denote the center of a group G?
Z(G)

Define the Center of a Group
 The center of a group G is the subset of elements in G that commute with every element of G.
 i.e. Z(G) = {a in G  ax = xa for all x in G}
The center of a group G is a subgroup of G.

Define the Centralizer
 Let a be a fixed element of a group G. The centralizer of a in G is the set of all elements in G that commute with a. Denoted C(a).
 i.e. C(a) = {g in G  ga = ag}
The centralizer is a subgroup

What does it mean to say a group is Cyclic?
 A group is cyclic such that there exists an element a in G such that G = {a^{n } n is an element of Z}
 denoted <a>

What is a Generator?
An element a is a generator of G when a^{n} = G

