Year: 2025
Type: article
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.
Source: Theoretical and Natural Science
Author: Muyao Wang
Institution: Xi’an Jiaotong-Liverpool University
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W4415182055
FWCI:
Cites:
Cited by:
Related to: