Definition of Groups
Binary Operation 
Let G be a set. A binary operation on G is a function that assigns each ordered pair of elements of G an element of G. 
Group 
Let G be a set together with a binary operation (usually called multiplication) that assigns to each ordered pair (a, b) of elements of G an element in G denoted by ab. We say G is a group under this operation if the following three properties are satisfied. 
Properties to Satisfy (Group) 
1. Closure 2. Associativity 3. Identity 4. Inverses 
Abelian group 
If a group has the property that ab = ba for every pair of elements a and b, we say the group is Abelian. 
Associativity 
The operation is associative; that is, (ab)c = a(bc) for all a, b, c in G. 
Identity 
There is an element e (called the identity) in G such that ae = ea = a for all a in G. 
Inverses 
For each element a in G, there is an element b in G (called an inverse of a) such that ab = ba = e. 
Modular Arithmetic 
When a = qn + r, where q is the quotient and r is the remainder upon dividing a by n, we write a mod n = r. 


Elementary Properties of Groups
Theorem 2.1 Uniqueness of the Identity 
In a group G, there is only one identity element. 
Theorem 2.2 Cancellation 
In a group G, the right and left cancellation laws hold; that is, ba = ca implies b = c, and ab = ac implies b = c. 
Theorem 2.3 Uniqueness of Inverses 
For each element a in a group G, there is a unique element b in G such that ab = ba = e. 
Theorem 2.4 SocksShoes Property 
For group elements a and b, (ab)^{1} = b^{1}a^{1}. 
Subgroup Tests
OneStep Subgroup Test 
Let G be a group and H a nonempty subset of G. If ab^{1} is in H whenever a and b are in H, then H is a subgroup of G. (In additive notation, if a  b is in H whenever a and b are in H, then H is a subgroup of G.) 
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 (H is closed under the operation), and a^{1} is in H whenever a is in H (H is closed under taking inverses), then H is a subgroup of G. 
Theorem 3.3 Finite Subgroup Test 
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. 


Examples of Subgroups
Theorem 3.4 <a> Is a Subgroup 
Let G be a group, and let a be any element of G. Then, <a> is a subgroup of G. 
Center of a Group 
The center, Z(G), of a group G is the subset of elements in G that commute with every element of G. In symbols, Z(G) = {a ∈ G  ax = xa for all x in G}. 
Theorem 3.5 Center Is a Subgroup 
The center of a group G is a subgroup of G. 
Centralizer of a in G 
Let a be a fixed element of a group G. The centralizer of a in G, C(a), is the set of all elements in G that commute with a. In symbols, C(a) = {g ∈ G  ga = ag}. 
Theorem 3.6 C(a) Is a Subgroup 
For each a in a group G, the centralizer of a is a subgroup of G. 
Terminology and Notation
Order of a Group 
The number of elements of a group (finite or infinite) is called its order. We will use G to denote the order of G. 
Order of an Element 
The order of an element g in a group G is the smallest positive integer n such that g^{n} = e. (In additive notation, this would be ng = 0.) If no such integer exists, we say that g has infinite order. The order of an element g is denoted by g. 
Subgroup 
If a subset H of a group G is itself a group under the operation of G, we say that H is a subgroup of G. 
