Proof and Applications of Lagrange’s Theorem in Deriving Fermat’s Little Theorem and Euler’s Theorem

Siyi Liu

doi:10.54254/2753-8818/2025.DL27546

Theoretical and Natural ScienceOpen access

Proof and Applications of Lagrange’s Theorem in Deriving Fermat’s Little Theorem and Euler’s Theorem

Research Article

Open Access

Proof and Applications of Lagrange’s Theorem in Deriving Fermat’s Little Theorem and Euler’s Theorem

Siyi Liu ^1*

¹ Raffles Institution

^*Corresponding author: 26YLIUS158W@student.ri.edu.sg

Published on 2 October 2025

TNS Vol.132

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

The group theory, as one of the cornerstones of the modern algebra, has a profound historical trajectory that reflects the evolution of the mathematical thought. This comprehensive paper analyses the historical development of the group theory and provides an overview of the interconnectedness of the several key theorems in the group theory: The Lagrange’s Theorem, the Fermat’s Little Theorem and the Euler’s Theorem. This paper begins by establishing the modern group-theoretical framework within the Lagrange’s Theorem on the link between the order of groups and that of its subgroups. Then, an extension onto other related theorems are provided. In all, this paper is highly interlinking among the ideas in group theory. Ultimately, this study not only demonstrates the beauty of mathematical interconnections but also highlights their continuing relevance for the modern applications, showing how the classical results remain relevant to guide contemporary explorations in algebra, number theory, and related disciplines.

Keywords:

Group theory, Lagrange’s Theorem, Fermat’s Little Theorem, Euler’s Theorem

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References

[1]. Kleiner, I. (1986). The evolution of group theory: A brief survey. Mathematics Magazine, 59(4), 195-215.

[2]. Hobbs, M., & Mallory, E. (2025). A Biography of Vilhjalmur Stefansson, Canadian Arctic Explorer (Vancouver: UBC Press, 1986); Gísli Pálsson, Travelling Passions: The Hidden Life of Vilhjalmur Stefansson, trans. Keneva Kunz (Winnipeg: University of Manitoba Press, 2005); and Janice Cavell and Jeff Noakes, Acts of Occupation: Canada and Arctic Sovereignty, 1918–25 (Vancouver: UBC Press, 2010). 20 On Stefansson's anthropological fieldwork in northern Canada, see Gísli Pálsson, ed., Writing. A Cold Colonialism: Modern Exploration and the Canadian North, 275.

[3]. Roth, R. L. (2001). A history of Lagrange's theorem on groups. Mathematics Magazine, 74(2), 99-108.

[4]. Lienert, C. (2023). Lagrange’s Proof of Wilson’s Theorem—and More!.

[5]. Joseph Vandehey. 2018. Lagrange’s Theorem for Continued Fractions on the Heisenberg Group.

[6]. Hassanzadeh, M. (2019). Lagrange's theorem for Hom-groups.

[7]. Armstrong, M. A. (1988). Lagrange’s theorem. In Groups and Symmetry (pp. 57-60). New York, NY: Springer New York.

[8]. Johnson, W. (1983). A note on Lagrange's theorem. The American Mathematical Monthly, 90(2), 132-133.

References

[1]. Kleiner, I. (1986). The evolution of group theory: A brief survey. Mathematics Magazine, 59(4), 195-215.

[3]. Roth, R. L. (2001). A history of Lagrange's theorem on groups. Mathematics Magazine, 74(2), 99-108.

[4]. Lienert, C. (2023). Lagrange’s Proof of Wilson’s Theorem—and More!.

[5]. Joseph Vandehey. 2018. Lagrange’s Theorem for Continued Fractions on the Heisenberg Group.

[6]. Hassanzadeh, M. (2019). Lagrange's theorem for Hom-groups.

[7]. Armstrong, M. A. (1988). Lagrange’s theorem. In Groups and Symmetry (pp. 57-60). New York, NY: Springer New York.

[8]. Johnson, W. (1983). A note on Lagrange's theorem. The American Mathematical Monthly, 90(2), 132-133.

Cite this article

Liu,S. (2025). Proof and Applications of Lagrange’s Theorem in Deriving Fermat’s Little Theorem and Euler’s Theorem. Theoretical and Natural Science,132,60-64.

Data availability

The datasets used and/or analyzed during the current study will be available from the authors upon reasonable request.

About volume

Volume title: Proceedings of CONF-APMM 2025 Symposium: Simulation and Theory of Differential-Integral Equation in Applied Physics

ISBN: 978-1-80590-305-5(Print) / 978-1-80590-306-2(Online)

Editor: Marwan Omar, Shuxia Zhao

Conference website: https://www.confapmm.org/dalian.html

Conference date: 27 September 2025

Series: Theoretical and Natural Science

Volume number: Vol.132

ISSN: 2753-8818(Print) / 2753-8826(Online)

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