Higman’s PORC conjecture for groups of prime-power order

Establish that, for each fixed positive integer k, the number of isomorphism types of groups of order p^k is polynomial in p on each residue class.

Background

The paper places its enumeration of groups of cubefree order within the broader problem of determining group numbers. In this context, it recalls Higman’s PORC Conjecture, which concerns the dependence on the prime p of the number of groups of order pk for fixed k. The conjecture is presented as a general unresolved conjectural framework rather than as a result proved in the paper.

References

For larger powers of $p$, asymptotic formulas exist, and Higman's PORC Conjecture claims that, for a fixed $k$, the number of groups of order $pk$ is a function of $p$ that is polynomial on residue classes.

— The number of groups of cubefree order  (2608.18815 - Dietrich et al., 19 Aug 2026) in Section 1, Introduction