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

Galois is the first to develop group theory. Group theory is considered to be a powerful formal method for analyzing abstract and physical sciences.The study of groups arose early in the nineteenth century in connection with the solution of equations.
 
Theory of abstract group plays an prominent role in present day mathematics and science. One of the most important intuitive ideas in mathematics and science is symmetry. Groups can describe symmetry and it explains to some extent why groups arise so frequently.
Geometric group theory is the study of groups as geometric objects. It is a rich field lying at a juncture between algebra and topology. Important idea in group theory finitely generated groups themselves as geometric objects. A simplest way of regarding a group as a geometric object is through cayley graph. A cayley graph of G is a graph whose vertices are precisely the elements of G. where the vertices are described by the rule that for elements x, y of G there is an edge labeled by the generator s_{i} originating at x and terminating at y provided xs$_{i}$ = y.
Combinatorial group theory is the study of groups defined in terms of permutations that is by means of generators and relations. Foundations of combinatorical group theory was led by  Walther von Dyck in the early 1880s.

It is a concept of a presentation of a group by generators and relations.
Usage of Combinatorial group theory is in geometric topology. It also contains algorithmically insoluble problems and the classical burn side problem.
Some of the problems to understand group theory are given below.

Example 1: Find the tables for addition mod 5 and multiplication mod 6.

Solution:
First we will find the table for addition mod 5 and later multiplication mod 6.
Addition mod 5
+$_{5}$
 0
 1 2
 3
 4
 0  0  1  2  3 4
 1  1  2  3  4  0
 2  2  3  4  0  1
 3  3  4  0  1  2
 4  4  0  1  2  3

Multiplication mod 6.
*$_{6}$
 0
 1 2
3
4
 5
 0  0
 0  0  0  0  0
 1  0  1  2  3  4  5
 2  0  2  4  0  2  4
 3  0  3  0  3  0  3
 4  0  4  2  0  4  2
 5  0  5  4  3  2  1

Example 2: On the set of natural numbers suppose operation * is defined by a * b = $\frac{a}{a+b}$. Check whether * is a binary operation.

solution: Given a * b = $\frac{a}{a+b}$

Let 3, 4 $\in$ to N. Then 3 * 4 = $\frac{3}{3+ 4}$ = $\frac{3}{7}$ $\notin$N
Therefore * is not a binary operation on N.

Example 3: On the set of rational numbers if * is defined by a * b = $\frac{ab}{7}$. Check whether * is associative.

Solution: Let a, b, c be any three rational numbers.

a * (b * c) = a * $\frac{bc}{7}$

 = $\frac{abc}{49}$

(a * b) * c = $\frac{ab}{7}$
* c

 = $\frac{abc}{49}$

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

Therefore * is associative.

Example 4: Show that G = {1, 2, 3} is not a group under multiplication mod 5.

Solution: The table of multiplication mod 5 is as follows.

*$_{5}$
1    
2      3  
 1  1  2  3
 2  2  4  1
 3  3  1  4

 The entries in the table are not the same as those in G.

Therefore the closure axiom is not satisfied.

Hence (G, *$_{5}$) is not a group.