Simple proofs of some generalizations of the Wilson’s theorem

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.