Graham Higman’s PORC Conjecture

Abstract

We survey the history of Graham Higman’s PORC conjecture concerning the form of the function f(pn) enumerating the number of groups of order pn. The conjecture is that for a fixed n there is a finite set of polynomials in p, g1(p), g2(p),…,gk(p), and a positive integer N, such that for each prime p, f(pn)=gi(p) for some i (1≤i≤k) with the choice of i depending on the residue class of p modulo N. We describe some properties of a group recently discovered by Marcus du Sautoy which has major implications for the PORC conjecture.