Simple proofs of some generalizations of the Wilson's theorem (Q482407)

Wilson's theorem which asserts that \((p-1)!\equiv -1\pmod p\) when \(p\) is a prime is generalized in the following way. If \(G\) is a finite abelian group then \(\prod_{g\in G} g=a\) if \(a\) is the only element of order \(2\) in \(G\), and the above product is \(1\) otherwise. This result is specialized to the case when \(G\) is the group of invertible elements modulo \(n\), where \(n\) is a positive integer. The last section reproves the known fact that the subgroup of the group of invertible elements modulo \(n\) consisting of the residue classes modulo \(n\) coprime to \(n\) of order at most \(2\) is cyclic if and only if there is a primitive root modulo \(n\).