A group is a set equipped with a binary operation that satisfies the following four properties:
Closure:
For any two elements , is also in .
Associativity:
For any three elements , .
Identity Element:
There exists an element such that for every element , .
Inverse Element:
For every element , there exists an element such that , where is the identity element.
Examples: non-zero real numbers
Let (the set of all non-zero real numbers) and define the operation as multiplication.
Closure:
The product of any two non-zero real numbers is also a non-zero real number.
Associativity:
Multiplication is associative: .
Identity Element:
The identity element is , because for any non-zero real number , .
Inverse Element:
For each non-zero real number , its inverse is , because .
Thus, is a group.
