Probabilistic construction of some extremal $p$-groups

Abstract

A $p$-group $G$ is called *ab-maximal* if $|H:H^{\prime }|<|G:G^{\prime }|$ 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 $|G:G^{\prime }|\ge p^3|G^{\prime }|$, while if $G$ is $d$-maximal and $p\ne 2$ then $|G:G^{\prime }|\ge p^2|G^{\prime }|$. Answering questions of González-Sánchez--Klopsch and Lisi--Sabatini, for all $p$ we construct infinitely many ab-maximal $p$-groups of class $2$ with $|G:G^{\prime }|=p^3|G^{\prime }|$, and infinitely many $d$-maximal $p$-groups of class $2$ with $|G:G^{\prime }|=p^2|G^{\prime }|$. 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 $|G:G^{\prime }|=p^n$ and $|G^{\prime }|=p^{n-3}$ that are ab-maximal is close to $1/e$.

15 pp. To appear in J. Algebra