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.
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.
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.
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.
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.