Simple proofs of some generalizations of the Wilson’s theorem

Jan Górowski; Adam Łomnicki

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica (2014)

  • Volume: 13, page 7-14
  • ISSN: 2300-133X

Abstract

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In this paper a remarkable simple proof of the Gauss’s generalization of the Wilson’s theorem is given. The proof is based on properties of a subgroup generated by element of order 2 of a finite abelian group. Some conditions equivalent to the cyclicity of (Φ(n), ·n), where n > 2 is an integer are presented, in particular, a condition for the existence of the unique element of order 2 in such a group.

How to cite

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Jan Górowski, and Adam Łomnicki. "Simple proofs of some generalizations of the Wilson’s theorem." Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 13 (2014): 7-14. <http://eudml.org/doc/268695>.

@article{JanGórowski2014,
abstract = {In this paper a remarkable simple proof of the Gauss’s generalization of the Wilson’s theorem is given. The proof is based on properties of a subgroup generated by element of order 2 of a finite abelian group. Some conditions equivalent to the cyclicity of (Φ(n), ·n), where n > 2 is an integer are presented, in particular, a condition for the existence of the unique element of order 2 in such a group.},
author = {Jan Górowski, Adam Łomnicki},
journal = {Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica},
keywords = {groups; congruences},
language = {eng},
pages = {7-14},
title = {Simple proofs of some generalizations of the Wilson’s theorem},
url = {http://eudml.org/doc/268695},
volume = {13},
year = {2014},
}

TY - JOUR
AU - Jan Górowski
AU - Adam Łomnicki
TI - Simple proofs of some generalizations of the Wilson’s theorem
JO - Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
PY - 2014
VL - 13
SP - 7
EP - 14
AB - In this paper a remarkable simple proof of the Gauss’s generalization of the Wilson’s theorem is given. The proof is based on properties of a subgroup generated by element of order 2 of a finite abelian group. Some conditions equivalent to the cyclicity of (Φ(n), ·n), where n > 2 is an integer are presented, in particular, a condition for the existence of the unique element of order 2 in such a group.
LA - eng
KW - groups; congruences
UR - http://eudml.org/doc/268695
ER -

References

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  1. [1] Lin Cong, Zhipeng Li, On Wilson’s theorem and Polignac conjecture, Math. Medley 32 (2005), 11-16. (arXiv:math/0408018v1). Cited on 7. 
  2. [2] J.B. Cosgrave, K. Dilcher, Extensions of the Gauss-Wilson theorem, Integers 8 (2008), A39, 15pp. Cited on 7 and 13. 
  3. [3] M. Hassani, M. Momeni-Pour, Euler type generalization of Wilson’s theorem, arXiv:math/0605705v1 28 May, 2006. Cited on 10. 
  4. [4] G.A. Miller, A new proof of the generalized Wilson’s theorem, Ann. of Math. (2) 4 (1903), 188-190. Cited on 7. 

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