cocenter

The term "cocenter" is a concept in group theory, a branch of abstract algebra. It is defined as the set of all elements in a group that commute with every element of the group. In other words, the cocenter of a group G is the set of all elements g in G such that for every h in G, the equation gh = hg holds true.

The cocenter of a group is always a subgroup of the group. It is also known as the centralizer of the group in its automorphism group. The cocenter is a useful concept in the study of group extensions and is closely related to the concept of the center of a group.

The cocenter of a group can be used to determine certain properties of the group. For example, if the cocenter of a group is trivial (i.e., it contains only the identity element), then the group is said to be cocentral. Cocentral groups have some interesting properties, such as being isomorphic to their automorphism groups.

The cocenter of a group can also be used to study the structure of the group. For example, the cocenter of a group can be used to determine the structure of the group's automorphism group. This can be useful in the study of group extensions and in the classification of groups.

In summary, the cocenter of a group is a subgroup that consists of all elements that commute with every element of the group. It is a useful concept in the study of group extensions and the structure of groups.