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 e_G 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.

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:
   • - 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 e_G 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.