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

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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

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##### Solution Summary

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

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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 ...

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