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 conclude that

xN{x^{  - 1}} = N\,\,\,\,\forall x \in G

Theorem 2: A subgroup N of a group G is a normal subgroup of Gif an 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. It 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 belongs 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 disjoint 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 subgroup of G.

 

Theorem 3: A subgroup N of a group G is 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. 

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