Group Theory

Group theory is often used in mathematics as a starting point for the study of many algebraic structures, such as a set of numbers along with its addition and multiplication. Because group theory is also useful for studying symmetry in nature and abstract systems, it has many applications in physics and chemistry.
We build up our definition of group step by step, from Semigroup to Monoid to Group.
A semigroup is simply a set together with an associative binary operation. Only associative is required, nothing else.
Equipping Semigroup with identity element, we get Monoid.
Adding inversion operation and its axioms to Monoid, we get Group.