Probabilistic construction of some extremal $p$-groups

TL;DR

This paper constructs infinite families of extremal $p$-groups of class 2 with specific subgroup index properties using probabilistic methods, answering open questions in group theory.

Contribution

It introduces a probabilistic construction of ab-maximal and $d$-maximal $p$-groups of class 2 with prescribed subgroup index ratios, resolving prior open problems.

Findings

Constructed infinitely many ab-maximal $p$-groups with $|G:G'|=p^3|G'|$

Constructed infinitely many $d$-maximal $p$-groups with $|G:G'|=p^2|G'|$

Proportion of such groups close to $1/e$ in the probabilistic model

Abstract

A $p$-group $G$ is called *ab-maximal* if $|H : H'| < |G:G'|$ for every proper subgroup $H$ of $G$. Similarly, $G$ is called *$d$-maximal* if $d(H) < d(G)$ for every proper subgroup $H$ of $G$, where $d(H)$ is the minimal number of generators of $H$. If $G$ is ab-maximal then $|G:G'| \ge p^3 |G'|$, while if $G$ is $d$-maximal and $p \ne 2$ then $|G:G'| \ge p^2 |G'|$. Answering questions of Gonz\'alez-S\'anchez--Klopsch and Lisi--Sabatini, for all $p$ we construct infinitely many ab-maximal $p$-groups of class $2$ with $|G:G'| = p^3 |G'|$, and infinitely many $d$-maximal $p$-groups of class $2$ with $|G:G'| = p^2 |G'|$. The construction is probabilistic and based on the degeneracy of random alternating bilinear maps on subspaces. It is notable however that in the ab-maximal case we do not have a high-probability result but rather in a suitable sense the proportion of class-$2$ groups with $|G:G'| = p^n$ and $|G'| = p^{n-3}$ that are ab-maximal is close to $1/e$.