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John Wilson (1741-1793) was a well-known English mathematician in his time, whose legacy lives on in his eponymous result, Wilson’s Theorem. To recall, this is the statement that an integer n >1 is prime if and only if

(n-1)! \equiv -1 \mod n.

(The “if” part is trivial.) As is the case for many historical results, Wilson’s Theorem was not proven by Wilson. Instead, it was Joseph Lagrange who provided the first proof.

The proof, as we see it today, might be phrased as follows:

Proof: Suppose that p is prime. Then each of the nonzero residues modulo p is a unit, so that (p-1)! represents the product over all units in \mathbb{Z}/p\mathbb{Z}. If

a \not\equiv a^{-1} \mod p,   i.e.    a^2 \not\equiv 1 \mod p,

then a and its inverse each show up in our list of units. We cancel out such terms in pairs, and conclude that

\displaystyle (p-1)! \equiv \prod_{a^2 \equiv 1 (p)} a \mod p.

We have a^2 \equiv 1 \mod p if and only if p \mid (a-1)(a+1), which by primality of p forces p \mid (a-1) or p \mid (a+1). In other words, a \equiv \pm 1 \mod p. It follows that

(p-1)! \equiv 1(-1) \equiv -1 \mod p.

\square

When n is composite, the direct translation of Wilson’s problem gives

(n-1)! \equiv 0 \mod n.

The problem, here, is that we’ve multiplied a number of zero divisors together, which can be avoided by only multiplying across the units, U_n, of \mathbb{Z}/n\mathbb{Z}. In this post, we consider the product

\displaystyle \prod_{a \in U_n} a \mod n,

determine its value, give credit to Gauss for doing so over two centuries ago, and discuss a few generalizations.

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In 1832, Galois introduced the concept of normal subgroups, and proved that the groups A_n (for n>4) and \mathrm{PSL}_2(\mathbb{F}_q) (for q>4) were simple, i.e. admit no non-trivial (proper) normal subgroups.  In 1892, Hölder asked for a classification of all finite simple groups, and the final classification of finite simple groups in 2004 by Aschbacher and Smith’s resolution of the quasithin case thus resolves an open problem over a century in the making. (Note: the great majority of the classification theorem concluded in the 1980’s.  A computational error was identified and resolved in 2008.)

One immediate consequence of the classification of finite simple groups (CFSG) is that the set S given by

S:=\{n \in \mathbb{N} : \text{there exists a simple group of order } n\}

has natural density 0.  Yet it seems unlikely that this result requires the tens of thousands of pages currently involved in the proof of the CFSG, and for the rest of this article we seek to bound the density of S using arithmetic methods.

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