Group

Any set with an operation (A, *) is called a group if and only if it satisfies the following four properties:

Closure Property

A set 'S' with respect to any operator * is said to satisfy closure property if ∀ a, b ∈ S, then a*b ∈ S.

If a set 'S' with respect to any operator * satisfy closure property, then it is known as algebraic structure.

A set 'S' with respect to any operator * is said to satisfy associative property if ∀ a, b, c ∈ S, then (a*b)*c = a*(b*c)

Note: Addition and multiplication operations holds associative property for any set.

If an algebraic structure satisfy associative property, then it is called semi-group.

A set 'S' wrt any operator * is said to satisfy identity property if ∀ a ∈ S, then is an element e ∈ S such that a*e = e*a = a.

Note: Identity element for addition is 0 (e = 0) and for multiplication is 1 (e = 1)for any set.

If a semi-group satisfy identity property, then it is called monoid.

A set 'S' wrt any operator * is said to satisfy inverse property if ∀ a ∈ S, there is an element a^-1 such that a*a^-1 = a^-1*a = e

If a monoid satisfy inverse property, then it is called a group.

A set 'S' wrt any operator * is said to satisfy commutative property if ∀ a, b ∈ S, then a*b = b*a.

If a group satisfy commutative property, then it is called an abelian group.