Group Theory – Basics
Group Theory is a branch of Algebra that study properties, operations & different aspects of the group. Such as, Group axioms, abelian group (commutative), Order of Group, etc..
Group Theory:
Definition (Group):
A nonempty element ‘G’ forms a ‘Group’ if in G, If we are defining a binary operation such that
 Closure Property: a,b∈G implies that a.b∈G.
 Associative Property: a,b,c∈G implies that a.(b.c)=(a.b).c
 Existence of unique identity: There exist an element e∈G s.t. a.e=e.a=a. for all a∈G.
 Existence of unique inverse: There exist an element a^{1}∈G s.t. a^{1}.e=e.a^{1}=1. for all a^{1}∈G.
Abelian Group:
A group G is ‘abelian’ or ‘commutative’ group if ∀ a,b∈G ⇒ a.b = b.a.
A group satisfying Abelian or Commutative property is obviously the Abelian or Commutative Group.
Order of a Group:
In Group Theory,Elements containing in the group, is nothing but the Order of a group.
Furthermore,
Order of group clearly inform us about the elements present in the group. i.e. The Elements contained in the Group.
We consider a example which will clear our ideas of Group theory
Example:
Assume that a structure containing integers {—, 2,1,0,1,2,—} we denote it by G i.e. G={—,2,1,0,1,2,—} . We define a binary operation addition(+) under which we are proving that G is an Abelian Group.
Proof:
In this example, at first, we prove the given algebraic structure G is a group furthermore, we prove it to be abelian.
G={—, 2,1,0,1,2,—} under operation addition(+)
Group axioms

Closure Property :
Here, we check addition of the elements in G.
Since, we already know the fact that addition of integers is always an integer.
a+b∈G ∀a,b∈G
So, obviously,G satisfies the Closure property.

Associative Property :
Same as above, we use the fact that addition of integers is associative.
surely,we say Associative property Satisfied.

Existence of Identity :
Clearly, ‘0’ is the additive identity in integers. Furthermore, ‘0’ exist in G.
∀ 0∈G s.t. 0+a=a+0=a
So, we can say a unique Identity exist in G.

Existence of inverse :
For any integer a∈G, there is (a)∈G such that
a+(a)=(a)+a=0, which is identity
As a result, we can say inverse exist in G.
All the Group axioms are satisfied. So, G is a group under operation addition.
At last, we prove that, the commutative property in integers

Commutative property :
The addition of integers is always commutative, likely, in this case Our group G is also commutative.
Finally, We can say, all the five properties satisfied. So, G is an abelian group under addition.
This is about the groups. We further discuss about types of group & the properties of groups in detail..