Concepts, Proof, and Applications of Lagrange’s Theorem

Abstract

The theory of groups and their representation, as a branch of mathematics, is a powerful tool for dealing with physical systems with certain symmetries. By using group theory method, many properties of the system can be directly understood qualitatively, complex calculations can be simplified, and the development trend of physical processes can be predicted. Lagrange’s theorem is a principle in Group Theory which is seen as the expansion of Euler’s theorem in number theory. It is considered an important concept to prove further complex theories in Group Theory. This study reviews these definitions and characteristics of subgroups and cosets, and then provides proofs for them. The main goal of the paper is demonstrating Lagrange’s Theorem that states that every quadratic irrationality has a periodic continued fraction. The structure of a Dirichlet group emerges from these properties of the unit group in an ordered environment. Additionally, the author shows how to use Gauss's reduction to calculate n-th roots of two-dimensional matrices method. Finally, the author will provide evidence for solutions to Lagrange’s Theorem.