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

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.


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

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.

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.


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 nondegenerate 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.