{"paper":{"title":"Maximal linear groups induced on the Frattini quotient of a $p$-group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alice C. Niemeyer, John Bamberg, Luke Morgan, S. P. Glasby","submitted_at":"2016-03-17T08:13:21Z","abstract_excerpt":"Let $p>3$ be a prime. For each maximal subgroup $H\\leqslant\\mathrm{GL}(d,p)$ with $|H| \\geqslant p^{3d+1}$, we construct a $d$-generator finite $p$-group $G$ with the property that $\\mathrm{Aut}(G)$ induces $H$ on the Frattini quotient $G/\\Phi(G)$ and $|G| \\leqslant p^{\\frac{d^4}{2}}$. A significant feature of this construction is that $|G|$ is very small compared to $|H|$, shedding new light upon a celebrated result of Bryant and Kov\\'acs. The groups $G$ that we exhibit have exponent $p$, and of all such groups $G$ with the desired action of $H$ on $G/\\Phi(G)$, the construction yields groups "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.05384","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}