Simple proofs of some generalizations of the Wilson’s theorem

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.