Quotient Groups

Definition: If G is a group and N is a normal subgroup of group G, then the set G|N of all cosets of N in G is a group with respect to multiplication of cosets. It is called the quotient group or factor group of G by N. The identity element of the quotient group G|N by N.

Theorem: The set of all cosets of a normal subgroup is a group with respect to multiplication of complexes as the composition.

Proof:
LetN be a normal subgroup of a group G. Since N is normal in G, therefore each right coset will be equal to the corresponding left coset.

Thus there is no distinction between right and left cosets and we shall call them simply as cosets. Let G|N be the collection of all cosets of N in G, i.e. let
G|N = \left\{ {Na:a \in G} \right\}

Closure Property: Let a,b \in G, then

\begin{gathered} \left( {Na} \right)\left( {Nb} \right) = N\left( {aN} \right)b \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = N\left( {Na} \right)b \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = NNab = Nab \\ \end{gathered}

Since ab \in G, therefore Nab is also a coset of N in G. So Nab \in G|N. Thus G|N is closed with respect to coset multiplication.

Associatively: Let a,b,c \in G. Then Na,Nb,Nc \in G|N. We have

\begin{gathered} Na\left[ {\left( {Nb} \right)\left( {Nc} \right)} \right] = Na\left( {Nbc} \right) \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = Na\left( {bc} \right) = N\left( {ab} \right)c \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left( {Nab} \right)Nc \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \left[ {\left( {Na} \right)\left( {Nb} \right)} \right]Nc \\ \end{gathered}

Thus the product in G|N satisfies the associative law.

Existence of Identity: We have N = Ne \in G|N. Also if Na is any element of G|N, then

\begin{gathered} N\left( {Na} \right) = \left( {Ne} \right)\left( {Na} \right) = Nea = Na \\ \left( {Na} \right)N = \left( {Na} \right)\left( {Ne} \right) = Nae = Na \\ \end{gathered}

Therefore the coset N is the identity element.

Existence of Inverse: Let Na \in G|N, then N{a^{ - 1}} \in G|N. We have

\begin{gathered} \left( {Na} \right)\left( {N{a^{ - 1}}} \right) = Na{a^{ - 1}} = Ne = N \\ \left( {N{a^{ - 1}}} \right)\left( {Na} \right) = N{a^{ - 1}}a = Ne = N \\ \end{gathered}

Therefore the coset N{a^{ - 1}} is the inverse of Na. Thus each element of G|N possesses inverse.

Hence G|N is a group with respect to product of cosets.