The Proof of Lagrange Theorem and Its Applications

Muyao Wang

doi:10.54254/2753-8818/2025.DL27731

Theoretical and Natural ScienceOpen access

The Proof of Lagrange Theorem and Its Applications

Research Article

Open Access

The Proof of Lagrange Theorem and Its Applications

Muyao Wang ^1*

¹ Xi’an Jiaotong-Liverpool University

^*Corresponding author: Muyao.Wang23@student.xjtlu.edu.cn

Published on 14 October 2025

TNS Vol.142

ISSN (Print): 2753-8826

ISSN (Online): 2753-8818

ISBN (Print): 978-1-80590-305-5

ISBN (Online): 978-1-80590-306-2

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Abstract

This paper explores Lagrange’s Theorem, a foundational result in abstract algebra that establishes a connection between the orders of a group and its subgroups. Initially introduced by Joseph Lagrange in the 18th century, the theorem asserts that the order of any subgroup divides the order of the entire group. This investigation begins with essential concepts of group theory, including cosets and bijections, leading to a rigorous proof of Lagrange’s Theorem. The paper also highlights significant implications of the theorem, such as its role in deriving Wilson’s Theorem and Fermat’s Little Theorem, both of which proves pivotal in algebraic theory. Furthermore, the applications of Lagrange’s Theorem in modern cryptography, particularly in the RSA public-key cryptosystem, are discussed, illustrating its relevance in contemporary mathematical practices. Despite its profound impact, there is no guarantee of the existence of subgroups for every divisor by the theorem, a limitation addressed by Sylow’s Theorem. This paper concludes by emphasizing the enduring significance of Lagrange’s Theorem in linking abstract algebra to practical applications and suggests avenues for future research in Galois theory and advanced cryptographic methods.

Keywords:

Group theory, Lagrange Theorem, RSA

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