# 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 $G$.

ProofLet $N$ 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.