Applications of Group Theory

Abstract

Groups play a fundamental role in Algebra: many algebraic structures, including rings, fields, and modules, can be seen as formed by adding new operations and axioms based on groups. Researchers often use group theory to explain many kinds of phenomena. In recent years, group theory has been introduced into crystallography to further explore the macroscopic symmetry of crystals from a mathematical point of view. In this paper, the applications of group theory in crystallography and magic cubic will be discussed. Basic definitions and models of these fields are demonstrated. A finite group is a group with a finite number of elements, which are the important contents of group theory. Besides, this paper proves that n−1 elements in a n order group can completely decide the nth element and gives a method of the nth element in a commutative group of order n. The analysis suggests that the research method of group theory has an important influence on other subjects.