Graham Higman's PORC theorem

TL;DR

This paper revisits Higman's PORC theorem, providing a simplified proof that incorporates recent algorithmic developments, clarifying the theorem's complex general result about the enumeration of p-groups.

Contribution

The paper offers a new, clearer proof of Higman's general PORC theorem, integrating recent computational insights and simplifying its complex original formulation.

Findings

Simplified proof of Higman's general PORC theorem

Algorithms for computing PORC formulas in special cases

Enhanced understanding of the theorem's key ideas

Abstract

Graham Higman published two important papers in 1960. In the first of these papers he proved that for any positive integer $n$ the number of groups of order $p^{n}$ is bounded by a polynomial in $p$, and he formulated his famous PORC conjecture about the form of the function $f(p^{n})$ giving the number of groups of order $p^{n}$. In the second of these two papers he proved that the function giving the number of $p$-class two groups of order $p^{n}$ is PORC. He established this result as a corollary to a very general result about vector spaces acted on by the general linear group. This theorem takes over a page to state, and is so general that it is hard to see what is going on. Higman's proof of this general theorem contains several new ideas and is quite hard to follow. However in the last few years several authors have developed and implemented algorithms for computing Higman's PORC formulae in special cases of his general theorem. These algorithms give perspective on what are the key points in Higman's proof, and also simplify parts of the proof. In this note I give a proof of Higman's general theorem written in the light of these recent developments.