Higman's PORC conjecture for a family of groups

TL;DR

This paper proves that for a fixed order exponent n, the number of groups with order p^n and a central Frattini subgroup varies with p as a PORC function, extending Higman's earlier work.

Contribution

It establishes that the count of such groups is a PORC function of p for fixed n, generalizing Higman's result.

Findings

Number of groups with order p^n and central Frattini subgroup is PORC in p.

Extends Higman's theorem to a broader class of groups.

Provides a polynomial-residue class characterization of group counts.

Abstract

We prove that the number of groups of order $p^n$ whose Frattini subgroup is central is for fixed $n$ a PORC (`polynomial on residue classes') function of $p$. This extends a result of G. Higman.