
Definition 1.4
A nonempty set G equipped with an operation * on it is said to form a group under that operation if the operation obeys the following laws, called the group axioms:
 1. Closure: for any a,b E G, we have a*b E G
 2. Associativity: For any a, b, c E G, we have a*(b*c)=(a*b)*c
 3. Identity: There exists an element e E G such that for all a E G we have a*e=e*a=a. Such an element e E G is called an Identity in G.
 4. Inverse: For each a E G there exists an element a^1 E G such that a*a^1=a^1*a=e. Such an element a^1 E G is called an inverse of a in G.

Proposition 1.18
Basic group properties: For any group G:
 1. The identity element of G is unique
 2. For each a E G, the inverse a^1 is unique
 3. For any a E G, (a^1)^1=a
 4. For any a, b E G, (ab)^1=b^1a^1
 5. for any a, b E G, the equations ax=b and ya=b have unique solutions or, in other words, the left and right cancellation laws hold

Theorem 1: Subgroup test
A nonempty subset H of a group G is a subgroup of G IFF the following condition holds:
 For every a, b E H, ab^1 E H
 For addition
 For every a, b E H, ab E H

Theorem 2: Subgroup test 2
A nonempty subset H of a group G is a subgroup of G IFF the following conditions hold:
 1. Closure: For every a, b E H, ab E H
 2. inverse: For every b E H, b^1 E H

Definition 2.13
Let G be a group and a E G. Then we define
 <a>={a^nn E Z}
 If the group operation is written as addition,
 <a>={na n E Z}

Definition 2.19
Let G be a group and a E G. The order a of a in G is?
 the least positive integer n such that a^n=e, of infinitie if there is no such n.
 If the group operation is written as addition, the condition would be written na=0

Definition 2.23
Let G be any group, Then the center of G, denoted Z(G), consists of?
The elements of G that commute with every element of G. In other words:
 Z(G)={x E G  xy=yx, Vy E G}
 Note: ey=y=ye for all y E G, so E E Z(G), and the center is a nonempty subset of G.

Theorem 2.24:
The Center Z(G) of a group is?
a subgroup of G.

Definition 2.26
Let G be a group and a E G. Then the centralizer of a in G, denoted by C_{G}(a), is?
 The set of all elements of G that commute with a. In other words:

 C_{G}(a)={y E G  ay =ya}
 Note tha for any a E G, we have Z(G) < C_{G}(a). In other words, the center is contained in the centralizer of any element.

Definition 3.4
A group G is called cyclic if?
 There exists an element a E G such that
 G=<a>. Any such element is called a generator of G.

Theorem 3.9:
Let G be a group and a E G. Then for all i,j E Z, we have:
 1. If a has infinite order, then a^i=a^j IFF i=j
 2. If a has finite order a=n, then a^i=a^j IFF n divides ij

Theorem 3.14
Let G=<a> be a cyclic group with generator a of order G=a=n. Then?
for any elemn a^s E G we have a^n=n/gcd(n,s)

Theorem 3.21
Every subgroup of a cyclic group G is?
Cyclic

Theorem 3.24
Let G=<a> be a cyclic group of order n. Then:
 1. the order H of any subgroup H of G is a divisor of n=G
 2. For each positive integer d that divides n there exists a unique subgroup of order d, the subgroup H=<a^^{n/d}>

