In 1968, Peter M. Neumann, Charles C. Sims and James Wiegold published in the *Journal of the London Mathematical Society* a paper with a title that would make any mathematician jealous: “Counterexamples to a theorem of Cauchy”.

Cauchy is best known for his work in analysis, which (with Weierstrass and others) put the subject on a sound footing not dependent on infinitesimals and similarly philosophically dubious notions. His theorems on complex analysis are at the heart of this beautiful subject. But there is more to him than that. He and Poisson were the Academicians who destroyed the career of Galois by losing his memoir. And Cauchy can also claim to be one of the founding fathers of group theory, with a stream of papers. One of his theorems is the famous result that, if a prime *p* divides the order of a finite group *G*, then *G* contains an element of order *p*: a precursor of Sylow’s Theorem, arguably the most important theorem in finite group theory.

The theorem in question here is a theorem of group theory. Let *G* be a group of permutations of a finite set *X*. We say that *G* is primitive if there is no non-trivial partition of *X* preserved by *G*. (The trivial partitions are the partition into singletons and the partition with just one part.) Also, we say that *G* is doubly transitive if any two points of *X* can be mapped to any other two (in either order) by some permutation in *G*. The degree of *G* is the number of elements in *X*.

With this out of the way we can state Cauchy’s “theorem”:

If

Gis a primitive permutation group whose degree is a prime plus one, thenGis doubly transitive.

A group theory book I used as a student, by W. R. Scott, mentioned this theorem, and included case-by-case proofs for a few small primes.

The paper of Neumann, Sims and Wiegold observes the following. Let *S* be a non-abelian finite simple group. Then the group *G*=*S*×*S* acts on the set *X*=*S* by the rule that the element (*g*,*h*) maps *x* to *g*^{-1}*xh*. The resulting permutation group is easily shown to be primitive but not doubly transitive; its degree is the order of the simple group *S*.

Now the orders of the first few simple groups are 60, 168, 360, 504, 660, 1092, 2448; and 59, 167, 359, 503, 659, 1091, and 2447 are all prime. So Cauchy’s Theorem is false.

But some questions remain:

- Are there infinitely many numbers of the form prime plus one for which there exists a primitive but not doubly transitive group of that degree? In other words, was Cauchy wrong infinitely often?
- Are there infinitely many orders of finite simple groups which have the form prime plus one? In other words, did Neumann
*et al.*provide infinitely many counterexamples?

The second question needs no knowledge of group theory to answer it; it is purely a question of number theory. Pick up any book on finite simple groups, and you will find a table of the orders of the finite simple groups. Often the expressions involve a prime power *q*. For example, if *q* is an odd prime power greater than 3, then the group PSL(2,*q*) is a simple group of order *q*(*q*+1)(*q*–1)/2. These simple groups are the most prolific, accounting for all but one of the short list above. So a particular question would be: are there infinitely many odd prime powers for which this number is prime plus one?

This seems to me to be a hard question, maybe comparable to the classical question whether *n*^{2}+1 is prime for infinitely many *n*.

In the opposite direction, I can prove is that there are infinitely many finite simple groups whose order is composite plus one. This might seem counter-intuitive in view of my short list above; but in fact only fifteen of the first thirty-six simple group orders are of the form prime plus one.

For example, 59 divides |PSL(2,5)|–1, from which it follows that 59 divides |PSL(2,*q*)|–1 whenever *q* is an odd prime power congruent to 5 (mod 59). Dirichlet’s theorem on primes in arithmetic progressions shows that infinitely many primes satisfy this congruence.

A more interesting question is whether Neumann *et al.* found all the counterexamples to Cauchy’s “theorem”. It turns out that they didn’t. Between 4 and 1000 there are 167 numbers of the form prime plus one; of these, five are orders of simple groups (60, 168, 360, 504, 660), and seven more are counterexamples to the “theorem” (68, 84, 102, 234, 462, 620, 840).

It would be interesting to know asymptotics of how many such numbers there are.

For the record, continuing the lists to 2500, we find two more orders of simple groups (1092 and 2448) and five more counterexamples to Cauchy’s theorem not of that form (1040, 1320, 1550, 1584 and 2162).

and just one more between 2500 and 4095, namely 4080 (the order of PSL(2,16)).

You’ll be able to tell from this that I’m not a mathematician, but could you explain , simply, to a layman, how there can be counter-examples to a theorem? I thought that once a theorem was proved there were no counter-examples. Hope this is an appropriate comment to make, apologies if it’s the wrong place to ask.

You are welcome to ask. The answer is that mathematicians are very casual in their use of the word “theorem”. We had “Fermat’s Last Theorem” which was actually a conjecture for a couple of hundred years.

Cauchy published a theorem whose statement I give in the post. But there was a mistake in his proof, and the supposed “theorem” is actually false. This happens more often than we might like to admit…

Usually what happens when someone points out a counterexample to a theorem is that the author is able to modify the statement of the theorem and find a revised proof to cover it. (This happened to me once.) This may simply involve adding the counterexample as a specific exception to the theorem, or strengthening the hypotheses to exclude it. But sometimes the proof is just wrong and cannot be repaired. That seems to be the case with Cauchy’s “theorem”. But even then, as I tried to show, there are often interesting things to say!

Curiously, just after posting this reply, I heard in a lecture (I am at a conference in Cochin, India) about a theorem published in 1992. The author “retracted” his theorem in 2000 and published a different result in its place. It turned out that the original theorem was correct and the replacement was wrong. (The author was a Russian cryptographer, and the person who found the mistake and also a new proof of the original theorem has been unable to contact him to tell him,)

Stephen Glasby has given me an even better infinite class of simple groups whose orders are of the form a composite plus one. These are the exceptional groups G

_{2}(q). We have|G

_{2}(q)|−1 = (q^{6}−q^{2}−1)(q^{8}−q^{6}+q^{4}−q^{2}+1),so none of these groups are counterexamples to Cauchy!