Definition, axioms and examples of a groups

This article contains information about what group is, its definition and examples.

Definition

The classic definition of a group is as follows:

A group is a set with a single binary operation that satisfies the following conditions:

1. 
2. 
3. 
4. 

Consider the following notes.

Note 1

It is important to understand that there is only one binary operation in the group. For example, we are familiar with real numbers. There are 4 operations with them: addition, subtraction, multiplication, and division.

Note 2

Sometimes, to make it easier to write, instead of just write , keeping in mind the operation defined in the group. Note that instead of can be any operation, not necessarily multiplication. However this operation is called multiplication.

Let's define a binary operation.

Binary operation is an operation that combines two elements of a set. For example, we are familiar with the binary plus operation in the set of integers 5+9. This operation combines two numbers 5 and 9 In contrast, there is also a unary operation that interacts with only one element of the set. An example of a unary operation is a unary minus in the set of integers. Negative numbers are denoted by a unary minus, such as −5.

Consider each axiom of the group in details.

Axioms

The first axiom of the group is called Closure. It is said that the group is closed relative to the operation defined in it. This property means that whatever elements of the group we consider, their product will always belong to the same group. For example - the product of these elements belongs to the group . This axiom can be understood as follows: . This means that after applying operation defined in the group to elements one will get element which belongs to the group .

The second axiom of the group is called: Associativity. This property means that there is no difference in the order in which the operation is applied to the group elements. For example, you can first apply the operation to elements , and then multiply the result on the right by . Or conversely, first apply the operation to elements , and then multiply the result on the left by .

The third axiom of the group is called: Existence of an identity element. This property means that there is an identity element in any group. This is an element that leaves any element of the group unchanged when combined with it. It is important to note that there is only one identity element in any group.

The fourth axiom of the group is called: The existence of an inverse element for any element of the group. This property means that any element of the group has an inverse one, which gives an identity element when combined with initial element.

Examples

1. - is a group of real numbers without zero with a multiplication operation.
   Let's check the implementation of the group's axioms:
   • Obviously, the result of multiplying any two real numbers is a real number.
   • The multiplication operation is associative.
   • The identity element in this group is the familiar real number one: .
   • For any real number there is an inverse fraction which belongs to the real numbers.
   This group also designated as follows: and is called the group of invertible elements of the set of real numbers or the multiplicative group of the field of real numbers. This name follows from the fact that all real numbers except 0 are invertible in this group.
2. - is a group of integers with an addition operation.
   Let's check the implementation of the group's axioms:
   • The result of multiplying any two integer numbers is an integer number.
   • The addition operation is associative.
   • The identity element in this group is an integer zero: 0.
   • For any integer there is an opposite number , which also belongs to the integers.
3. - is the group of all geometric vectors in space with the vector addition operation.
   These are familiar vectors in the rectangular cartesian coordinate system Oxyz. Remember that you can use the triangle or parallelogram rule to add two vectors. For adding three or more vectors, you must use the rule of the polygon.
   Let's check the implementation of the group's axioms:
   • The result of addition of any two vectors is a vector.
   • The vectors addition operation is associative.
   • The identity element in this group is the zero vector .
   • For any vector there is an opposite vector .