Quotient Group

A quotient group is the set of cosets of a normal subgroup of a group. Let N be a normal subgroup of group G. If x be any arbitrary element in G, then Nx is a right coset of N in G, and xN is a left coset of N in G. But as N is a normal group, by definition of normal group, the right coset of N in G will equal to the left coset of N in G, that is there is no difference between the left and right coset of a normal subgroup of a group.

This gives rise to an interesting question which is, “does the set of cosets form a group?”. Hence, the quotient group can be simply defined as the group of cosets of a normal subgroup of a group, with respect to the composition rule of the group.

Definition of Quotient Group

If G is a group and N is a normal subgroup of G, then the set G/N of all cosets of N in G is a group with respect to the composition of the group.

Thus, the elements of G/N are in form of cosets like Na₁, Na₂, Na₃, …where a₁, a₂, a₃,… are in G.

Let Nx and Ny be in G/N then we shall prove that (Nx)(Ny) is in G/N

(Nx)(Ny) = NxyN (since N is a normal subgroup.

Let, xy = z in G, as both x and y belong to G.

(Nx)(Ny) = NzN = NNz = Nz ∈ G/N (since NN = N)

Hence, G/N is closed with respect to the multiplication composition of the group.

Let Na, Nb, Nc are in G/N where a, b and c are in G. Then,

[(Na)(Nb)]Nc = N(ab)c = Na(bc) {since associativity holds within G}

= (Na)N(bc) = Na[N(b)N(c)]

Thus, G/N is associative as well.

Since N = Ne ∈ G/N, for any element Na in G/N, we have

(Na)N = (Na)(Ne) = N(ae) = Na = N(ea) = (Ne)(Na) = N(Na)

Where e is the identity element in G

Hence, N is an identity element in G/N.

If a is an element of G then a^-1 is also an element of G, hence Na and Na^-1 are elements of G/N such that

(Na)(Na^-1) = N(aa^-1) = Ne = N

Similarly, (Na^-1)(Na) = N.

Thus, for every element in G/N there exists an inverse of it in G/N.

With this, we prove that G/N is a group distinctly called the Quotient Group of G.

Example of a Quotient Group

Let G be the addition modulo group of 6, then G = {0, 1, 2, 3, 4, 5} and N = {0, 2} is a normal subgroup of G since G is an abelian group.

Now, G/N = { N+a | a is in G} = {{0, 2}, {1, 3}, {2, 4}} = { N+0, N+1, N+2}

N + 3 = {3, 5} which will be the same as N + 0.

Let us check whether G/N is group

Closure Property: N+1 and N+2 in G/N

(N + 1)+(N + 2) = N + N + ( 1 + 2) = N + 3 = N + 0 ∈ G/N

Thus G/N is closed with respect to addition modulo 6.

Associativity: (N+0)[(N+1)+(N+2)] = N + (0+1+2) = N + 3 ∈ G/N

[(N+0)(N+1)]+(N+2) = N + (0+1+2) = N + 3 = N + 0 ∈ G/N

Thus G/N is associative with respect to addition modulo 6.

Identity: Now N +(N + 2) = (N + 0) + (N + 2) = N + 2 = (N + 2) + (N + 0) = (N + 2) + N

Therefore N = N + 0 ∈ G/N is the identity element in G/N.

Inverse: Now, (N + 1)+(N + 2) = N + 3 = N + 0 = (N + 2)+(N + 1)

Thus, N+1 and N+2 are inverse elements of each other in G/N

Therefore, G/N is a Quotient group of G, which is also abelian, as G/N is commutative as well.

Properties of a Quotient Group

• The identity element of a quotient group is the normal subgroup itself.
• If N is a normal subgroup of G, the Order of G/N is equal to the order of G divided by the order of N.That is, |G/H| = |G|/|N|
• Quotient group of an abelian group is abelian, but the converse is not true.
• Every quotient group of a cyclic group is cyclic, but the opposite is not true.
• The quotient group G/G has correspondence to the trivial group, that is, a group with one element.
• The quotient group G/{e} has correspondence to the group itself.
• If G is nilpotent then so is the quotient group G/N.
• If G is solvable then the quotient group G/N is as well.

Related Articles:

• Group
• Linear Algebra
• Group Theory
• Abstract Algebra

Solved Examples on Quotient Group

Example 1:

Let G be the additive group of integers and N be the subgroup of G containing all the multiples of 3. The quotient group of G is given by G/N = { N + a | a is in G}. Find the order of G/N.

Solution:

Given G = {…-2, -1, 0, 1, 2, 3,…}

And N = {…, -6, -3, 0, 3, 6,…}

G/N = { N + a | a is in G}

then , N + 1 = {…, -5, -2, -1, 2, 5,…}

N + 2 = {…, -4, -1, 2, 5, 8,…}

Now a = 3b + c where b is in G and c = 0, 1, 2.

Therefore, N + a = N + (3b + c) = (N + 3b) + c = N + c

As 3b belong to N.

Thus, G/N = { N, N + 1, N + 2}

Now, Order of G/N = Index of N in G = Number of cosets of N in G = 3.

Example 2:

Let G = {1, -1, i, -i} be a multiplicative group and N = { 1, -1} be a subgroup of G. Find the number of elements in the quotient group of G.

Solution:

Clearly, G is abelian being a multiplicative group, then N is a normal subgroup.

The quotient group G/N = {Na | a is G}

N1 = { 1, -1} = N

N(-1) = { -1, 1} = N

Ni = { i, -i}

N(-i) = {-i, i} = Ni

G/N = {N, Ni}

Therefore, the number of elements in the quotient group of G is 2.