QUOTIENT GROUP

Quotient Group:
Let G be a group.

Let N be a normal subgroup.

Then the left coset space G/N is a group, where the group product is defined as:

         (aN)(bN) = (ab)N

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

It is proven to be a group in Quotient group is Group.

A quotient group is also known as a factor group.
In other words, We can say 
If H is a normal subgroup of a group G, then the group G/H of all the right cosets of H in G under the composition
         (Ha)(Hb) = Hab is called a quotient group or a factor group.

Note: If the composition in G/H is addition, then the composition in G/H is defined by
           (H + a) + (H + b) = H + (a + b).

Remark: If H is a normal subgroup of a finite group G, then G/H form a group of order O(G)/O(H).

Theorem: If H is a subgroup of an abelian group G, then the group G/H of all right cosets of H in G forms an abelian group under the composition defined by Ha.Hb = Hab.
Proof: If H is a subgroup of an abelian group G, then H is normal subgroup of G.
∴        G/H forms a quotient group.
Let Ha, Hb ∈ G/H      so that a,b ∈ G.
(Ha)(Hb) = Hab = Hba,  since G is abelian.   ∴   ab = ba
                  = (Hb)(Ha).
Hence G/H is an abelian group.

Converse: The converse of the above result is not true that is, the quotient group may be abelian even if G may not be abelian.

Theorem: Every quotient group of a cyclic group is cyclic.
Proof: Let G = <a> be a cyclic group generated by a.
∴ G is an abelian group.
∴ Each subgroup of G is normal subgroup.
Let h be any subgroup of G.
∴ H is a normal subgroup of G.
So G/H form a quotient group.
We prove that G/H is a cyclic group generated by Ha.
Let Hx ∈ G/H be arbitrary element, where x ∈ G.
∴ But G = <a>
∴ x = aⁿ for some integer n.
∴ Hx = Haⁿ = H a . a ... a (n times)
                       = Ha. Ha ... Ha (n times) 
                       = (Ha)ⁿ 
∴ Hx = (Ha)ⁿ , ∀ Hx ∈ G/H.
∴ G/H is a cyclic group generated by Ha.
So, each quotient group of a cyclic group is cyclic.

Converse: The converse of the above theorem may not be true.
i.e, quotient group may be cyclic even if the group may not be cyclic.

Example: If H be a normal subgroup of a group G and [G : H] = m, then show that for any x ∈ G, xⁿ ∈ H.
Sol. Since H is a normal subgroup of group G such that 
[G : H] = m.
∴ O(G/H) = m.
∴ ∀ xH ∈ G/H,    where x ∈ G, we have
      (xH)^m = H    [∵ If O(G) = n then aⁿ = e, ∀ a ∈ G]
      x^mH = H
⇒  x^m ∈ H.
Thus ∀ x ∈ G,   we have  x^m ∈ H.