Additive and multiplicative notation of a binary operation in a group, examples

The most common notations of a binary operation in a group are additive: + and multiplicative: •.

Additive notation of a binary operation

If in a group Group designation the binary operation binary operation in group is an addition operation, then this group is called an additive one and is denoted as Additive group designation.

Examples of additive groups

  1. Any ring or field is a group with an addition operation (the addition operation is taken from the ring) and is called an additive group of the ring or field.
    Consider examples of additive groups:
    • Additive group of integers - is the additive group of the ring of integers. The set of integers The set of integers is commutative associative ring with 1 with the familiar addition and multiplication operations.
    • Additive group of real numbers - is the additive group of the field of real numbers. The set of real numbers The set of real numbers is a field with the familiar operations of addition and multiplication of real numbers.
    • Additive group of complex numbers - is the additive group of the field of complex numbers. The set of complex numbers The set of complex numbers is a field with the familiar operations of addition and multiplication of complex numbers.
    • Additive group of rational numbers - is the additive group of the field of rational numbers. The set of rational numbers The set of rational numbers is a field with the familiar operations of addition and multiplication of fractions.
  2. Additive group of all geometric vectors in space - is the additive group of all geometric vectors in space.

Multiplicative notation of a binary operation

If in a group Group notation the binary operation binary operation in group is an multiplicative operation, then this group is called a multiplicative one and is denoted as Multiplicative group notation.

Note

There are also multiplicative groups of rings or fields. To consider examples of such groups, first consider the definitions of some terms.

Definition of an inverse element in a ring. Let Associative ring with 1 - is an associative ring with 1, then the element Invertible element of an associative ring with 1 is called invertible, if there is an element Inverse element b to element a such, that: definition of the inverse element , and b = a−1.

Notation. The set of all invertible elements of a ring Associative ring with 1 is denoted as The set of all invertible elements of a ring.

Corollary. In the A field , all elements except 0 are invertible elements. That is The set of all invertible elements of a field.

Examples of multiplicative groups

  1. The set of all elements except 0 (the identity element for addition) of the field field is a group with a multiplication operation (the multiplication operation is taken from the field). This group is called the multiplicative group of the field and is denoted as The multiplicative group of the field.
    Consider the examples of multiplicative groups of fields:
    • Multiplicative group of a field of real numbers - is the multiplicative group of the field of real numbers.
    • Multiplicative group of a field of complex numbers - is the multiplicative group of the field of complex numbers.
    • Multiplicative group of a field of rational numbers - is the multiplicative group of the field of rational numbers.
  2. The set of all invertible elements of an associative ring with 1 - Associative ring with 1 is the group with a multiplication operation (the multiplication operation is taken from the ring). This group is called the multiplicative group of the ring and is denoted by The multiplicative group of the ring.
    Consider the examples of multiplicative groups of rings:
    • Multiplicative group of the ring of integers - is the multiplicative group of the ring of integers. Note that in a ring of integers, only the numbers 1 and -1 are invertible numbers, because the invertible elements for integers are fractions, but fractions are not integers. For example, for the integer 5, the inverse element is the inverse fraction Inverse fraction to the number 5. This fraction is not an integer. Therefore there is no inverse element in integers for the integer 5. However, for the numbers 1 and −1 there are inverse elements - these are inverse fractions: Inverse fraction to the number 1 and Inverse fraction to the number -1. This fractions are integers. So, the invertible elements for integers 1 and −1 are themselves.
  3. General linear group - is the group of all invertible (nondegenerate) square matrices of size n×n with the matrix multiplication operation.
    This group is called the general linear group and is defined as: The definition of the general linear group. The matrix is called non-degenerate or invertible if its determinant is not equal to 0. The set of all real numbers - in parentheses means that the elements of the matrix X are real numbers. This group will be discussed in more details in the following articles.

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