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 G is a primitive permutation group whose degree is a prime plus one, then G is 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-1xh. 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 n2+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.