# Normalizer &amp; normal subgroup related

• iibewegung
In summary, the normalizer of a subgroup H in a group G is the set of all elements in G that commute with every element in H. It is denoted as N<sub>G</sub>(H) and is the largest subgroup of G in which H is a normal subgroup. The centralizer of H in G is a subgroup of the normalizer, and both play important roles in the study of group theory. Multiple normalizers can exist for a subgroup in a group, but their intersection will always be the centralizer of the subgroup. Normalizers can also be used to classify groups into different types, such as abelian, solvable, and simple groups.
iibewegung
[SOLVED] Normalizer &amp; normal subgroup related

Let $$H \subset G$$.
Why is $$H$$ a normal subgroup of its own normalizer in $$G$$?

Sorry for wasting space... I figured it out.
(wasn't reading the definition of normalizers carefully)

It was one of those things that one could attach "Clearly," at the front of the statement.

Unfortunately, this leads to another question...
How do I delete this thread?

Last edited:

## 1. What is a normalizer in a group?

The normalizer of a subgroup H in a group G is the set of all elements in G that commute with every element in H. It is denoted as NG(H). In other words, it is the largest subgroup of G in which H is a normal subgroup.

## 2. How is the normalizer related to the centralizer in a group?

The centralizer of a subgroup H in a group G is the set of all elements in G that commute with every element in H. This is similar to the definition of the normalizer, but the centralizer only requires elements to commute with elements in H, while the normalizer requires them to commute with every element in G. In other words, the centralizer is a subgroup of the normalizer.

## 3. What is the significance of normalizers in group theory?

Normalizers play an important role in the study of group theory as they help us understand the structure of a group. They can be used to determine whether a subgroup is normal in a group, and can also be used to construct new groups from existing ones.

## 4. Can a subgroup have more than one normalizer in a group?

Yes, a subgroup can have multiple normalizers in a group. This can occur when there are multiple subgroups that are normal in a group, and each of these subgroups have their own corresponding normalizers. However, the intersection of these normalizers will always be the centralizer of the subgroup.

## 5. How can normalizers be used to classify groups?

Normalizers can be used to classify groups into different types, such as abelian, solvable, and simple groups. For example, a group is abelian if and only if all of its subgroups are normal, and thus have normalizers equal to the entire group. Similarly, a group is simple if and only if its only normal subgroups are the trivial subgroup and the entire group, and thus have normalizers equal to themselves and the entire group.

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