Geodesic
Simple proofs of some generalizations of the Wilson’s theorem
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica, no. 13 (2014) 
Górowski, Jan; Łomnicki, Adam. Simple proofs of some generalizations of the Wilson’s theorem. Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica, no. 13 (2014). http://geodesic.mathdoc.fr/item/AUPCM_2014_13_a8/
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Voir la notice de l'article provenant de la source Library of Science

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.

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[2] J.B. Cosgrave, K. Dilcher, Extensions of the Gauss-Wilson theorem, Integers 8 (2008), A39, 15pp. Cited on 7 and 13.

[3] M. Hassani, M. Momeni-Pour, Euler type generalization of Wilson’s theorem, arXiv:math/0605705v1 28 May, 2006. Cited on 10.

[4] G.A. Miller, A new proof of the generalized Wilson’s theorem, Ann. of Math. (2) 4 (1903), 188-190. Cited on 7.