{"paper":{"title":"A characterization of powerful p-groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Amaia Zugadi-Reizabal, Jon Gonzalez-Sanchez","submitted_at":"2013-07-02T08:14:18Z","abstract_excerpt":"In [10] Benjamin Klopsch and Ilir Snopce posted the conjecture that for $p\\geq 3$ and $G$ a torsion-free pro-$p$ group $d(G)=\\dim (G)$ is a sufficient and necessary condition for the pro-$p$ group $G$ to be uniform. They pointed out that this follows from the more general question of whether for a finite $p$-group $d(G)=\\log_p(|\\Omega_1(G)|)$ is a sufficient and necessary condition for the group $G$ to be powerful. In this short note we will give a positive answer to this question for $p\\geq 5$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.0613","kind":"arxiv","version":1},"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"}