Quotient Group in Group Theory

Prerequisite : Knowledge of Groups, Cosets.

Introduction :
We can say that  “o” is the binary operation on set G if : G is an non-empty set & G * G = { (a,b) : a , b∈ G } and  o : G * G –> G. Here, aob denotes the image of ordered pair (a,b) under the function / operation o.
Example –  “+” is called a binary operation on G (any non-empty set ) if & only if : a+b ∈G ; ∀ a,b ∈G and a+b give the same result every time when added.

Algebraic Structure :
A non-empty set G equipped with 1/more binary operations is called algebraic structure.
Example – a. (N,+)  and b. (R, + , .), where N is a set of natural numbers & R is a set of real numbers. Here ‘ . ‘ (dot) specifies a multiplication operation.

GROUP :
An algebraic structure (G , o) where G is a non-empty set & ‘o’ is a binary operation defined on G is called a Group if the binary operation “o” satisfies the following properties –

Closure – a ∈ G ,b ∈ G => aob ∈ G ; ∀ a,b ∈ G 

1. Associativity – (aob)oc = ao(boc) ; ∀ a,b,c ∈ G. 
   
2. Identity Element – There exists e in G such that aoe = eoa = a ; ∀ a ∈ G (Example – For addition, identity is 0). 
   
3. Existence of Inverse – For each element a ∈ G ; there exists an inverse(a-1)∈ G such that : aoa-1 = a-1oa = e. 
   

Abelian Group : 
An algebraic structure (G , o) where G is a non-empty set & ‘o’ is a binary operation defined on G is called an abelian Group if it is a group (i.e. , it satisfies G1, G2, G3 & G4) and additionally satisfies 

Commutative - aob = boa ∀ a,b ∈ G

Normal Subgroup :
Let G be an abelian group & the composition in G has been denoted by multiplicity.
Let H be any subgroup of G. If x is an arbitrary element of G, the Hx is the right coset of H in G & xH is the left coset of H in G, then G is called a normal subgroup if –

Hx = xH ; ∀x ∈ G or
xhx-1 ∈ H ; ∀x ∈ G & h ∈ H

Quotient Group :
Let G be any group & let N be any normal Subgroup of G. If ‘a’ is an element of G , then aN is a left coset of N in G. Since N is normal in G, aN = Na ( left coset = right coset).
We can say that Na is the coset of N in G. 
G/N denotes the set of all the cosets of N in G.

Quotient/Factor Group = G/N = {Na ; a ∈ G } = {aN ; a ∈ G} (As aN = Na)

If G is a group & N is a normal subgroup of G, then, the sets G/N of all the cosets of N in G is a group with respect to multiplication of cosets in G/N. It is called the quotient / factor group of G by N.
Sometimes it is called ‘Residue class of G modulo N’.
If the composition in the group is addition, ‘+’, then G/H is defined as :

Quotient/Factor Group = G/N = {N+a ; a ∈ G } = {a+N ; a ∈ G} (As a+N = N+a)

NOTE –  The identity element of G/N is N.
Example 1 – Consider the group G with addition modulo 6 where G = {0, 1, 2, 3, 4, 5}. Let N = {0, 3),
then the quotient/ factor group is :
G/N = { aN ; a ∈ G } = { a{0,3} ; a ∈ {0, 1, 2, 3, 4, 5}}
= {0{0,3}, 1{0,3}, 2{0,3}, 3{0,3}, 4{0,3}, 5{0,3} }
= { {(0+0) mod6 , (0+3) mod6 }, { (1+0) mod6 , (1+3) mod6 } , { (2+0) mod6 , (2+3) mod6 } , { (3+0) mod6 , (3+3) mod6 }, { (4+0) mod6 , (4+3) mod6 }, { (5+0) mod6 , (5+3) mod6 } }
= {{0,3}, {1,4}, {2,5}, {3,0}, {4,1}, {5,2} }
=  {{0,3}, {1,4}, {2,5}}

Example 2 – Let G = {1, -1, i, -i } and H = {1, -1}; H is the normal subgroup of G in binary operation ‘,’ . What will be the quotient group; G/H?
G/N = { aN ; a ∈ G } = {a{1,-1} ; a ∈ {1,-1,i,-i}
= {1.{1,-1}, -1.{1,-1}, i{1,-1}, -i.{1,-1}}
={{1.1,1.-1}, {-1.1,-1.-1}, {i.1, i.-1}, {-i.1, -i.-1}}
={{1,-1}, {-1,1}, {i,-i}, {-i,i}}
={ {1,-1}, {i,-i}}
In other words, we can say that if G is a group & N is a normal subgroup of G, then G/N of all the cosets of N in G together with a binary composition defined by :

NaNb = Nab ; where Na ∈ G/N, Nb ∈ G/N is a group.

G/N is called the quotient group of G by N.

Properties of Quotient/ Factor group :

1. If N is a normal subgroup of a finite group G, then –
   O(G/N) = O(G)/O(N), where : O(G/N) => No of distinct right / left cosets of N in G.
2. If N is a normal subgroup of a finite group G such that the  index of N in G is prime, the factor group G/N is cyclic.
3. The factor group of an abelian group is abelian, but the converse is not true.
4. Every factor group of a cyclic group is cyclic but the converse is not true.