Today I attended (remotely) a nice talk by Gareth Jones in the Ural Workshop on Group Theory and Combinatorics, about prime powers in permutation group theory and polynomials taking prime values in number theory.
I will give just one example result from the first part. For a prime p, a p-complement in a group G is a subgroup of order coprime to p whose index is a power of p. Philip Hall showed that a finite soluble group contains a unique conjugacy class of p-complements for all primes p. The new result is:
Theorem Given a prime p and a finite group G, there is at most one p-complement in G, up to automorphisms of G.
On the second topic, Gareth mentioned two conjectures which are well out of reach of number theorists at present. Let fi, for i = 1,…,k, be integer polynomials. We are interested in integers t for which all the values fi(t) are prime, and in particular, whether there are infinitely many such. There are a couple of obvious necessary conditions for this: all the polynomials should be non-constant and irreducible, with positive leading coefficients; and no positive integer should divide the value of the product of the polynomials for all possible arguments. Schinzel’s hypothesis asserts that these necessary conditions are in fact sufficient, and the Bateman–Horn conjecture goes further by providing a conjectured asymptotic for the number of values of t up to some given limit x.
It is clear that the first conjecture would show that there are infinitely many twin primes, and infinitely many Sophie Germain primes, while the second would give an asymptotic estimate of their number.
Gareth gave an application to permutation groups; I will give another.
I have mentioned here before the “theorem” of Cauchy to which Peter Neumann, Charles Sims and James Wiegold gave counterexamples. Cauchy had claimed that a primitive permutation group of degree a prime plus one is necessarily doubly transitive. The counterexamples are obtained by choosing a finite simple group S whose order is a prime plus one, and letting S×S act on S where one factor acts on the left and the other on the right. I have discussed this here.
It is not known (at least to me) whether there are infinitely many counterexamples to Cauchy’s theorem, and in particular whether the NSW construction gives infinitely many. But the conjectures just described would imply that there are, if I am not mistaken. Let us take the simple group PSL(2,p), where p is an odd prime. Putting p = 2t+1, we see that we require values of t for which the polynomials 2t+1 and 2t(t+1)(2t+1)−1 to take prime values.
Probably similar results hold for some other families of simple groups of Lie type. However, in the comments to the previous post, I noted that Stephen Glasby pointed out to me that it will not work for the groups G2(q); the order of this group minus 1 is a reducible polynomial, with factors of degrees 6 and 8.
About Peter Cameron
I count all the things that need to be counted.
