In any course on Elementary Number Theory you meet Wilson’s Theorem. This says that for any prime we have that .

How do we prove this? Well I have reshaped the usual proof in order to generalise.

My claim: For any finite Abelian group, the product of all elements is equal to the product of 2-torsion elements (i.e. the self-inverse ones).

This is easy to see…every element in the product has to have an inverse (we are in a group) and either a given element is self-inverse or not. Thus when taking the product as a whole everything that is not self-inverse gets inverted (if we jumble up the product)…leaving only those elements that are self-inverse.

Now we may prove Wilson’s theorem. The non-zero integers mod form a finite group under multiplication. The product of all elements is the same as . By the above this product is the same as the product of 2-torsion elements. These ones will correspond to solutions of . Solving we find that are the only 2-torsion elements…hence . QED

In a similar vein we find that for any and odd prime then discarding the multiples of p from gives us something which is also .

Maybe other finite groups will provide other Wilson’s Theorems?

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