NTAG Seminar: The number and nature of subgroups of the symmetric group

The symmetries of any object are described by a group, so it is natural to ask: What does a random group look like? This talk will start with a brief survey of how we might go about counting various algebraic structures. We'll then go on to see what a random group might be, in various different contexts. A symmetric group on some set Omega is the group of all permutations of Omega, under composition of functions. Every group arises as a subgroup of some symmetric group, so fully understanding the symmetric group means understanding all groups. An elementary argument shows that there are at least 2^{n^2/16} subgroups of a symmetric group on n points, and it was conjectured by Pyber in 1993 that up to lower order error terms this is also an upper bound. The same year, Kantor conjectured that a random subgroup of the symmetric group is nilpotent. This talk will present a proof of one of these conjectures, and a disproof of the other. New results in this talk are joint w/ Gareth Tracey.