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Primes dividing binomial coefficients, invariable generation of finite simple groups, and noncontractibility of coset posets
Algebra & Discrete Mathematics| Speaker: | John Shareshian, Washington University in St. Louis |
| Location: | 2112 MSB |
| Start time: | Mon, Mar 9 2026, 11:00AM |
Description
I will discuss joint work with Bob Guralnick (USC) and Russ Woodroofe (University of Primorska).
A theorem of Kummer implies that unless n is a prime power, the greatest common divisor of the nontrivial bionmial coefficients {{n} \choose {k}}, 0<k<n, is 1. With this in mind, we aim to partition the set of nontrivial binomial coefficients in as few subsets as possible so that the elements of each such subset have gcd larger than 1. We know of no n thast requires more than two subsets.
After discussing what we know about the binomial coefficient problem, I will explain how this problem arose in our study of invariable generation of finite simple groups by two subgroups sastisfying certain conditions. (Two subsets H,K of a group G generate G invariably if <a^{-1}Ha,b^{-1}Kb>=G for all a,b in G.).
Using results on invariable generation along with Smith Theory, we have shown that the order complex of the poset all all cosets of all proper subgroups of an arbitrary finite group G, ordered by setwise inclusion, has nontrivial rational homology and is therfore not contractible. This answers a question raised by Ken Brown.