# Wilson's Theorem : Cyclic Groups and Order of an Element

13.a) If G={g1,g1,....,gr} is an abelian group, show that g1,g2....gr equals the product of the elements of order 2.

b) Prove Wilson's Theorem: If p is a prime then (p-1)! R (-1)(modp)

note: R is a equivalence relation

https://brainmass.com/math/finite-element-method/wilson-s-theorem-cyclic-groups-and-order-of-an-element-14143

#### Solution Preview

Proof:

(a) G={g1,g2,...,gr} is an abelian group. Suppose A is the set of all the elements of order 2 in G. Let A={a1,a2,...,as}.

for an element x in A, we have x^2=1 or x=x^(-1). But ...

#### Solution Summary

Wilson's theorem is proven using abelian groups. The proof is concise.

$2.19