Simple proofs of some generalizations of the Wilson's theorem

Jan Górowski, Adam Łomnicki

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), 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.

Keywords


Wilson's theorem, finite abelian group, cyclicity

Mathematics Subject Classification (2010)


20K01, 11A07

References


Cong, Lin and Zhipeng Li. "On Wilson’s theorem and Polignac conjecture." Math. Medley 32 (2005): 11–16.

Cosgrave, J.B. and K. Dilcher. "Extensions of the Gauss-Wilson theorem." Integers 8 (2008), A39, 15pp.

Hassani, M. and M. Momeni-Pour. "Euler type generalization of Wilson’s theorem." 28 May, 2006.

Miller, G.A. "A new proof of the generalized Wilson’s theorem." Ann. of Math. (2) 4 (1903): 188–190.


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