Gareth Tracey

The number of subgroups of a finite group

In his 1993 Ann. Math. paper, L. Pyber proves a beautiful upper bound on the
number of isomorphism classes of finite groups of order n. Key to his proof is an upper bound
on the number of subgroups of the symmetric group Sym(n). This upper bound however, is not
sharp, and Pyber makes a conjecture as to what the “correct” bound should be. In this talk,
we will discuss some recent progress on the conjecture, and some consequences in the theory of
random subgroups of a finite group. Joint work with Colva Roney-Dougal.