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 non-empty element ‘G’  forms a ‘Group’ if in G, If we are defining a binary operation such that-

  1. Closure Property:  a,b∈G implies that a.b∈G.
  2. Associative Property: a,b,c∈G implies that  a.(b.c)=(a.b).c
  3. Existence of unique identity: There exist an element e∈G s.t. a.e=e.a=a. for all a∈G.
  4. 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..