# 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 the binary operation is an addition operation, then this group is called an additive one and is denoted as .

• In additive group, the result of applying the addition operation to elements is called the sum and is denoted as .
• A identity element eG in an additive group is called zero and is denoted as 0.
• The inverse element of the g element is denoted as -g and is called the opposite element.

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.
• - is the additive group of the ring of integers. The set of integers is commutative associative ring with 1 with the familiar addition and multiplication operations.
• - is the additive group of the field of real numbers. The set of real numbers is a field with the familiar operations of addition and multiplication of real numbers.
• - is the additive group of the field of complex numbers. The set of complex numbers is a field with the familiar operations of addition and multiplication of complex numbers.
• - is the additive group of the field of rational numbers. The set of rational numbers is a field with the familiar operations of addition and multiplication of fractions.
2. - is the additive group of all geometric vectors in space.

## Multiplicative notation of a binary operation

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

• In multiplicative group, the result of applying the multiplication operation to elements is called the product and is denoted as or as .
• An identity element eG in a multiplicative group is called unit and is denoted as 1.
• The inverse element to the element g is denoted as g-1 and is also called the inverse element.

## 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 - is an associative ring with 1, then the element is called invertible, if there is an element such, that: , and b = a−1.

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

Corollary. In the , all elements except 0 are invertible elements. That is .

## Examples of multiplicative groups

1. The set of all elements except 0 (the identity element for addition) of the 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 .
Consider the examples of multiplicative groups of fields:
• - is the multiplicative group of the field of real numbers.
• - is the multiplicative group of the field of complex numbers.
• - is the multiplicative group of the field of rational numbers.
2. The set of all invertible elements of an 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 .
Consider the examples of multiplicative groups of rings:
• - 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 . 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: and . This fractions are integers. So, the invertible elements for integers 1 and −1 are themselves.
3. - 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 matrix is called non-degenerate or invertible if its determinant is not equal to 0. - 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.