A group ( S , ⊕ ) is a set S together with a binary operation ⊕ defined on S for which the following properties hold.

1. Closure: For all a , b ∈ S , we have a ⊕ b ∈ S.
2. Identity: There is an element e ∈ S , called the identity of the group, such that e ⊕ a = a ⊕ e = a, for all a ∈ S.
3. Associativity: For all a , b , c ∈ S, we have ( a ⊕ b) ⊕ c = a ⊕ ( b ⊕c).
4. Inverses: For each a ∈ S , there exists a unique element b ∈ S , called the inverse of a , such that a ⊕ b = b ⊕ a = e .

As an example, consider the familiar group ( Z , +) of the integers Z under the operation of addition: 0 is the identity, and the inverse of a is - a .

Abelian group :

If a group ( S , ⊕ ) satisfies the commutative law a ⊕ b = b ⊕ a, for all a , b ∈ S , then it is an abelian group .

 

The groups defined by modular addition and multiplication

First we define the congruence notation ≡ as follows:

If a , b ∈ Z then we say a ≡b modulo n if ∃ p , q , r ∈ Z such that a = pn + r and b = qn + r .

We will denote a mod n as [a]_n

We can form two finite abelian groups by using addition and multiplication modulo n , where n is a positive integer. These groups are based on the equivalence classes of the integers modulo n

a ≡ a ' (mod n ) and b ≡b ' (mod n ), then

a + b ≡ a ' + b ' (mod n ) ,

ab ≡ a ' b ' (mod n ) .