# 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.

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Group theory

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