Abstract: Fix an integer n>1. It follows directly from a theorem of Kummer that the greatest common divisor of the members of the set BC(n) nontrivial binomial coefficients nC1,nC2,...nC(n-1) is one unless n is a prime power. With this in mind, we define b(n) to be the smallest size of a set P of primes such that every member of BC(n) is divisible by at least one member of P. In joint work with Russ Woodroofe, we ask whether b(n) is at most two for every n. The question remains open.
I will discuss what we know about this question, and how we discovered it during our investigation of a problem raised by Ken Brown about certain topological spaces: Given a finite group G, let C(G) the set of all cosets of all proper subgroups of a finite group, partially ordered by containment. The order complex of C(G) is the simplicial complex whose k-dimensional faces are chains of size k+1 from C(G). We show that this order complex has nontrivial reduced homology in characteristic two, and is therefore not contractible.
If time permits, I will discuss also related work on invariable generation of simple groups, joint with Bob Guralnick and Russ Woodroofe.