In the eyes of many mathematicians who study group theory, that is, abstract group theory, mathematicians care about the operational relationship between elements, that is, the structure of a group, without considering the specific meaning of the elements of the group.

For example, according to Kelley theorem, any group is isomorphic to a permutation group composed of elements of the group. Therefore, especially for studying finite groups, it is an important problem to study permutation groups.

In mathematics and abstract algebra, group theory studies the algebraic structure called groups. Groups play a fundamental role in abstract algebra: many algebraic structures, including rings, fields and modules, can be regarded as being formed by adding new operations and axioms to groups. The concept of group theory appears in many branches of mathematics, and the research methods of group theory also have an important influence on other branches of abstract algebra.

The importance of group theory is also reflected in the research of physics and chemistry, because many different physical structures, such as crystal structure and hydrogen atom structure, can be modeled by group theory. Therefore, group theory and related group representation theory have many applications in physics and chemistry.