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^-1 denotes the inverse of element a∈ A under the operation *. In group theory the notation a^-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 aⁿ 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.