Normalizer of a Subgroup:

 Normalizer of a Subgroup:

 Let The set of those elements of which permute with is called normalizer of in Mathematically, it can be written as:

 

NG(H)={gH=Hg : g∈G}

Results:

1. Normalizer of a subgroup of a group is always a subgroup of 

 

2. In an abelian group 

 subgroup of 

3. For any subgroup of H of G Z(G) ≤NG(H)

 

4. Let H ≤G and |H |= 2  then CG(H)=NG(H)

 

 

5. Let G  be a finite group then a∈Z(G) and a∈NG(H)

 

6. Let H ≤ G | G| =p^n if p divides [G:H] then NG(H) ≠H

 

 7. For any a ∈ G , Z(G)= ⋂a ∈G NG(a)

 

 

8. For any group  ⟨e ⟩ = G=NG(G)

 

9. For any group G NG(Z (G) ) =G