Non-PORC behaviour of a class of descendant $p$-groups

TL;DR

This paper demonstrates that the count of immediate descendants of a specific class of $p$-groups does not follow a PORC pattern, challenging assumptions related to Higman's PORC conjecture.

Contribution

It proves that the number of immediate descendants of a certain $p$-group is not polynomial on residue classes, providing a counterexample relevant to Higman's conjecture.

Findings

Number of immediate descendants is not PORC.

Implications for Higman's PORC conjecture.

Counterexample in $p$-group enumeration.

Abstract

We prove that the number of immediate descendants of order $p^10$ of $G_p$ is not PORC (Polynomial On Residue Classes) where $G_p$ is the $p$-group of order $p^9$ defined by du Sautoy's nilpotent group encoding the elliptic curve $y^2=x^3-x$. This has important implications for Higman's PORC conjecture.