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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.05821 |
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| _version_ | 1866915357930815488 |
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| author | Eberhard, Sean Sabatini, Luca |
| author_facet | Eberhard, Sean Sabatini, Luca |
| contents | 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 González-Sánchez--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 $1/e$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05821 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Probabilistic construction of some extremal $p$-groups Eberhard, Sean Sabatini, Luca Group Theory 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 González-Sánchez--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 $1/e$. |
| title | Probabilistic construction of some extremal $p$-groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2502.05821 |