Group theory is a mathematical concept. 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.

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The concept of group originated from Evariste Galois's research on polynomial equations in 1930s. After getting contributions from other fields such as number theory and geometry, the concept of group was formed and firmly established around 1870. Modern group theory is a very active mathematics subject, which studies groups in its own way. In order to explore groups, mathematicians invented various concepts, and decomposed groups into smaller and better understood parts, such as permutation groups, subgroups, quotient groups, simple groups and so on.

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