Groupremains

Groupremains is a term used in the field of group theory, a branch of abstract algebra. It refers to the set of elements in a group that remain unchanged under a specific group action. This concept is particularly relevant in the study of group actions and permutation groups.

In formal terms, let G be a group acting on a set X. The groupremains of this action is the subset of X consisting of all elements x in X such that gx = x for all g in G. This subset is also known as the fixed point set or the set of invariants under the group action.

The study of groupremains is crucial in various areas of mathematics and its applications. For instance, in topology, the fixed point set of a continuous map can provide valuable information about the map's properties. In combinatorics, groupremains can help in understanding the symmetries of certain structures.

The concept of groupremains is closely related to other important notions in group theory, such as stabilizers and orbits. The stabilizer of an element x in X is the subgroup of G consisting of all elements g such that gx = x. The orbit of x is the set of all elements in X that can be obtained from x by applying elements of G. The groupremains is essentially the intersection of all orbits in X.

In summary, groupremains is a fundamental concept in group theory that plays a significant role in understanding group actions and their properties. Its study has applications in various areas of mathematics and its applications.