A combinatorial approach to wilson’s theorem for finite abelian groups

Document Type : Translation Paper

Authors

1 Department of Mathematics Education, Farhangian University, Tehran, Iran

2 Department of Mathematics, Babol Branch, Islamic Azad University, Babol, Iran

Abstract

This paper is a translation of the the following paper into Persian:
[Chase Saucier, A Combinatorial Approach to Wilson’s Theorem for Finite Abelian Groups, Mathematics Magazine, 91 no. 2 (2018) 97–102.]
 
 Here, we will obtain a few results in group theory without using the standard theorems of Lagrange and Cauchy, or even the basic algebraic concepts of subgroups, quotient groups, or homomorphisms. Instead, we use simple counting and involutive arguments. Recall that Wilson’s theorem says that $(p-1)!\equiv‎ -‎1~~(\mathrm{mod}‎ ~‎~p)$ if $p$ is prime (see [G. E. Andrews, Number Theory, Dover Publications, New York, 1994.]). Note that $(p-1)!$ is the product of all the units modulo $p$. We study this problem in a more general setting, and ask: Given an arbitrary fnite Abelian group $G$, what can be said of the product over all the elements of $G$? The answer is well-known, but diffcult to fnd in a standard textbook. We present our own approaches and give references to others’ approaches below
 
 

References

[1] G. E. Andrews, Number Theory, Dover Publications, New York, 1994.
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[3] P. L. Clark (n. d.), Wilson’s theorem:An algebraic approach, http://alpha.math.uga.edu/$\sim$pete/wilson-easy.pdf.
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[6] M. Isaacs, Algebra: A Graduate Course, CA: Cole Publishing Company, Pacific Grove, 1994.
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[8] G. A. Miller, A new proof of the generalized Wilson’s theorem, Ann. Math., 4 (1903) 188–190.
[9] O. Ore, Number Theory and Its History, McGraw-Hill Book Company, Inc, New York, 1948
Mathematics and Society
Volume 4, Issue 3 - Serial Number 3
September 2019
Pages 73-79

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History

  • Receive Date: 16 November 2019
  • Revise Date: 09 June 2020
  • Accept Date: 09 July 2020
  • Publish Date: 22 November 2019

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