In other words, the centralizer of H is the set of elements that leave any element of H unchanged when it is multiplied by any element of the group. Normalize...
In other words, the centralizer of H is the set of elements that leave any element of H unchanged when it is multiplied by any element of the group.
Normalizer:
A normalizer of a subgroup H is a subgroup of the group that contains the elements of H that commute with all elements of the group except for a finite subset of elements. In other words, the normalizer of H is the set of elements that leave all elements of H except for a finite number of elements unchanged when it is multiplied by any element of the group.
Here are some examples:
The normalizer of a cyclic group of order 4 is the entire group, since any element in the group can be written as a product of 4 elements of the group.
The center of a group G is the set of elements that commute with all elements of G, which is equal to G itself.
The center of a cyclic group of order 4 is the subgroup generated by the element 2, which consists of the elements 2, 4, 6, and 8.
These concepts are used in group theory to study the structure of subgroups and cyclic groups, as they help to determine the size and properties of these groups