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 The group designation is a set with a single binary operation The operation of the group that satisfies the following conditions:

  1. Axiom 1
  2. Axiom 2
  3. Axiom 3
  4. Axiom 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 The group designation just write The shorter group designation, keeping in mind the operation defined in the group. Note that instead of The operation of the group 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 The group designation we consider, their product will always belong to the same group. For example Product of two arbitrary elements of the group - the product of these elements belongs to the group The group designation. This axiom can be understood as follows: Closure. This means that after applying operation defined in the group to elements Two arbitrary group elements one will get The result of the product of two elements of the group, element that belongs to the group The group designation.

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

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 single 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 all elements of the group have a inverse element. An inverse element is an element that gives and indentity element when combined with that element.

Examples

  1. The group of real numbers without zero with a multiplication operation - 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: Действительное число единица.
    • For any real number: Any nonzero real number there is an inverse fraction: Inverse fraction that belongs to the real numbers.
    Also, this group is designated as follows: Multiplicative group of a field of real numbers 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 Set of real numbers except 0 are invertible in this group.
  2. The group of integers with a addition operation - 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 a integer number.
    • The addition operation is associative.
    • The identity element in this group is an integer zero: 0.
    • For any integer: Any integer there is an opposite number: The opposite of an integer, that belongs to the integer.
  3. The group of all geometric vectors in space with the vector addition operation - 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 adding any two vectors is a vector.
    • The vectors addition operation is associative.
    • The identity element in this group is the zero vector Zero vector.
    • For any vector Any vector in space there is an opposite vector: The opposite vector.

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