# Groups

A non-empty set A is called a **group** with respect to an internal binary operation *:A×A→A $$ *: \ A×A \rightarrow A $$ if it satisfies the following properties

**Closure**

if a∈A and b∈A then a*b∈A $$ a*b \in A \ \ \ \forall \ a,b \in A $$**Associative law**

given three elements a, b, c ∈ A, the binary operation * satisfies the associative property (a * b) * c = a * (b * c) $$ (a * b) * c = a * (b * c) \ \ \ \ \ \forall \ a,b,c \in A $$**Existence of identity element**

there is an element u ∈ A called identity element such that a * u = u * a = a for all elements a ∈ A $$ a * u = u*a = a \ \ \ \ \ \forall \ a \in A $$**Existence of inverse**

for each element a∈A there is an inverse element a^{-1}∈A such that a * a^{-1}= a^{-1}* a = u $$ a * a^{-1} = a^{-1} * a = u \ \ \ \ \ \forall \ a \in A $$

**The notation a**. In group theory the notation a^{-1}denotes the inverse of element a∈ A under the operation *^{-1}should not be confused with the inverse of multiplication. For example, if the binary operation * is addition +, the symbol a^{-1}is the inverse of addition -a. Only if the binary operation * is multiplication, the symbol a^{-1}is to be interpreted as the multiplication inverse.

To indicate the group we write (A, *) or A (*). In both cases the set and the binary operation of the group is indicated.

$$ (A,*) $$

A group can be composed of an infinite number of elements or a finite number of elements.

The order of a group (A,*) is the number of elements in the set A. It's denoted |A|.

A group is said to be **finite ( or finite order )** if it has finite number of elements.

A group of **infinite order** has infinite elements.

The order of an element a∈A is the least positive integer for which a^{n} is egual to the identity element (u) of group $$ a^n = u $$

A group is said to be **abelian** if the group operation is commutative.

$$ a * b = b*a \ \ \ \forall \ a,b \in A $$

It is called non-abelian if it is not commutative.

**Example**

The set of integers Z forms a group with respect to addition.

- The addition of three integers a, b, c ∈ Z satisfies the associative property. $$ (a + b) + c = a + (b + c) \ \ \ \ \ \forall \ a,b,c \in Z $$
- The identity element is 0. $$ a+0=0+a=a \ \ \ \ \ \forall \ a \in Z $$
- Each element a∈Z has an inverse element (-a)∈Z with respect to addition. $$ a+(-a)=(-a)+a=0 \ \ \ \ \ \forall \ a \in Z $$

Thus, (Z, +) is an __additive group__.