Theorems of Normal Subgroups

Theorem 1: A subgroup $$N$$ of a group $$G$$ is normal if and only if $$xN{x^{ – 1}} = N\,\,\,\,\forall x \in G$$.

Proof: Let $$xN{x^{ – 1}} = N\,\,\,\,\forall x \in G$$, then $$xN{x^{ – 1}} \subset N\,\,\,\,\forall x \in G$$. Therefore $$N$$ is a normal subgroup of $$G$$.

Conversely, let $$N$$ be a normal subgroup of $$G$$. Then
\[xN{x^{ – 1}} \subset N\,\,\,\,\forall x \in G\,\, – – – \left( i \right)\]

Also $$x \in G \Rightarrow {x^{ – 1}} \in G$$. Therefore we have
\[ {x^{ – 1}}N{\left( {{x^{ – 1}}} \right)^{ – 1}} \subset N\,\,\,\forall x \in G\]
\[ \Rightarrow {x^{ – 1}}Nx \subset N\,\,\,\forall x \in G\]
\[ \Rightarrow x\left( {{x^{ – 1}}nx} \right){x^{ – 1}} \subset xN{x^{ – 1}}\,\,\,\forall x \in G\]
\[ \Rightarrow N \subset {x^{ – 1}}Nx\,\,\,\forall x \in G\,\, – – – \left( {ii} \right)\]

From (i) and (ii) we can conclude that
\[xN{x^{ – 1}} = N\,\,\,\,\forall x \in G\]

Theorem 2: A subgroup $$N$$ of a group $$G$$ is a normal subgroup of $$G$$ if and only if each left coset of $$N$$ in $$G$$ is a right coset of $$N$$ in $$G$$.

Proof: Let $$N$$ be a normal subgroup of $$G$$ then \[xN{x^{ – 1}} = N\,\,\,\forall x \in G\]
\[ \Rightarrow \left( {xN{x^{ – 1}}} \right)x = Nx\,\,\,\forall x \in G\]
\[ \Rightarrow xN = Nx\,\,\,\forall x \in G\]
$$ \Rightarrow $$ each left coset $$xN$$ is the coset $$Nx$$

Conversely, let each left coset of $$N$$ in $$G$$ be a right coset of $$N$$ in $$G$$. This means that if $$x$$ is any element of $$G$$, then the left coset $$xN$$ is also a right coset. Now $$e \in N$$, and therefore $$xe = x \in xN$$. So $$x$$ must also belong to that right coset which is equal to left coset  $$xN$$. But $$x$$ is an element of the right coset $$Nx$$, and two right cosets are either disjointed or identical, i.e. if two right cosets contain one common element then they are identical. Therefore $$Nx$$ is the unique right coset which is equal to the left coset $$xN$$.

Therefore, we have
\[xN = Nx\,\,\,\forall x \in G\]
\[ \Rightarrow xN{x^{ – 1}} = Nx{x^{ – 1}}\,\,\,\forall x \in G\]
\[ \Rightarrow xN{x^{ – 1}} = N\,\,\,\forall x \in G\]
$$ \Rightarrow $$ $$N$$ is normal a subgroup of $$G$$.

Theorem 3: A subgroup $$N$$ of a group $$G$$ is a normal subgroup of $$G$$ if and only if the product of two right cosets of $$N$$ in $$G$$ is again a right coset of $$N$$ in $$G$$.

Theorem 4: The intersection of two normal subgroups of a group is a normal subgroup.