# Group Theory

In mathematics under groups there are various subtopics like abelian group, group and semigroup, modular groups, group of matrices etc. Groups are based on some axioms. Its axioms are short and natural. Axioms for groups give no obvious hint that anything like this exists.

Definition of a group is: A group is a set G together with a binary operation

(a,b) -> a * b; G * G -> G

satisfying the following axioms.

Associativity: $\forall$ a, b, c $\in$ G

( a * b) * c = a * ( b * c)

There exists an element e $\in$ G such that

a * e = a = e * a $\forall$ a $\in$ G.

For each a $\in$ G there exists an a' $\in$ G such that

a * a' = e = a' * a