This article contains information about what group is, its definition and examples.
The classic definition of a group is as follows:
A group is a set with a single binary operation that satisfies the following conditions:
Consider the following notes.
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.
Sometimes, to make it easier to write, instead of just write , keeping in mind the operation defined in the group. Note that instead of 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.
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 we consider, their product will always belong to the same group. For example - the product of these elements belongs to the group . This axiom can be understood as follows: . This means that after applying operation defined in the group to elements one will get element which belongs to the group .
The second axiom of the group is called: Associativity. This property means that there is no difference in the order in which the operation is applied to the group elements. For example, you can first apply the operation to elements , and then multiply the result on the right by . Or conversely, first apply the operation to elements , and then multiply the result on the left by .
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 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 any element of the group has an inverse one, which gives an identity element when combined with initial element.