What is the meaning of N(H) in subgroup notation?

In summary, N(H) or N_G(H) refers to the normalizer of H in G, which is the set of elements in G that map H to itself under conjugation. The task is to prove that either N_G(H) is equal to G, which would make H a normal subgroup, or N_G(H) is equal to H, which would make H a self-normalizing subgroup.
  • #1
Silviu
624
11
Hello! I have this problem:

If H is a subgroup of prime index in a finite group G, show that either H is a normal subgroup or N(H) = H.

What does N(H) means? I don't want a solution for the problem (at least not yet), I just want to know what that notation means. Thank you!
 
Physics news on Phys.org
  • #2
N(H) or better ##N_G(H)## is very likely the normalizer of ##H## in ##G##.
That is ##N_G(H)=\{g \in G \,\vert \, gHg^{-1} \subseteq H\}\,##.
The task here is to show that either ##N_G(H)=G## or ##N_G(H)=H\,##.
The first means a normal subgroup, the latter is called a self-normalizing subgroup.
 

What is "Notation N(H) for a subgroup"?

The notation N(H) for a subgroup refers to the normalizer of a subgroup H in a group G. It is the subgroup of G that contains all elements of G that commute with every element of H.

Why is the normalizer important in group theory?

The normalizer is important in group theory because it allows us to define a subgroup of G that is stable under conjugation by elements outside of H.

How is the normalizer different from the centralizer?

The normalizer and centralizer are both subgroups of G, but the normalizer contains elements that commute with every element of H, while the centralizer only contains elements that commute with every element of G.

What is the relationship between the normalizer and the subgroup H?

The normalizer of H is the largest subgroup of G in which H is a normal subgroup. In other words, the normalizer is the subgroup that contains all elements of G that preserve the structure of H.

Can the normalizer of H be equal to H itself?

Yes, the normalizer of H can be equal to H itself. This occurs when H is a normal subgroup of G, meaning that it is invariant under conjugation by elements of G.

Similar threads

MHB Can You Prove These Properties of Normal Subgroups in Group Theory?
    • Linear and Abstract Algebra
Replies
2
Views
1K
MHB Show that the groups are abelian
    • Linear and Abstract Algebra
Replies
1
Views
794
I Normalizer of a subgroup of prime index
    • Linear and Abstract Algebra
Replies
6
Views
2K
I Is G simple?Is G simple? A different approach using Sylow theorems
    • Linear and Abstract Algebra
Replies
5
Views
1K
MHB Proving N(H) is a Subgroup of G Containing H
    • Linear and Abstract Algebra
Replies
1
Views
1K
I Trying to get the point of some Group Theory Lemmas
    • Linear and Abstract Algebra
Replies
3
Views
899
MHB Proof: Every Subgroup of Cyclic $H$ is Normal in $G$
    • Linear and Abstract Algebra
Replies
12
Views
3K
MHB There is no proper subgroup
    • Linear and Abstract Algebra
Replies
1
Views
1K
MHB 4 part problem regarding homomorphisms
    • Linear and Abstract Algebra
Replies
1
Views
1K
POTW Groups of Prime Power Order
    • Math POTW for University Students
Replies
0
Views
114

Hot Threads

Recent Insights

Back
Top