Defintion of a group

Definition of a Group

A group is a mathematical structure that consists of a set of elements and a binary operation that combines any two of the elements to form a third element, subject to four conditions of operation.
Set of Elements: A group G must consist of a set of elements. The nature or type of these elements can vary widely.
Binary Operation: This operation takes any two elements in the set and combines them to produce a third element that must also belong to the set.

Closure: Closure refers to the property of a group that, for every pair of elements a and b in the group G, the result of the binary operation on a and b (denoted as a * b) will also be an element in the set that makes up group G.
Associativity: Associativity means that for every triple of elements (a, b, c) in the group G, the result of operating on a with the result of operating on b and c (i.e., (a * b) * c) is the same as the result of operating on a and b first and then operating on the result with c (i.e., a * (b * c)).
Identity Element: The identity element of a group G is an element that, when combined with any other element of the group through the group operation, leaves that element unchanged. The identity is usually denoted by “e” or “i”.
Inverse Element: For every element a in a group G, there is another element b in the group such that when the operation is carried out with a and b, the result is the group’s identity element. Here, b is the inverse of a.

Putting these four conditions together, a group is a set of elements equipped with a binary operation that combines any two elements to form a third, and which satisfies the conditions of closure, associativity, identity and inverse.
Understanding these basic properties and the definition of a group is fundamental for group theory, one of the major branches of abstract algebra, with many applications in mathematics and science alike.