{"paper":{"title":"Lower central words in finite $p$-groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Iker de las Heras, Marta Morigi","submitted_at":"2019-07-26T11:02:12Z","abstract_excerpt":"It is well known that the set of values of a lower central word in a group $G$ need not be a subgroup. For a fixed lower central word $\\gamma_r$ and for $p\\ge 5$, Guralnick showed that if $G$ is a finite $p$-group such that the verbal subgroup $\\gamma_r(G)$ is abelian and 2-generator, then $\\gamma_r(G)$ consists only of $\\gamma_r$-values. In this paper we extend this result, showing that the assumption that $\\gamma_r(G)$ is abelian can be dropped. Moreover, we show that the result remains true even if $p=3$. Finally, we prove that the analogous result for pro-$p$ groups is true."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.11479","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"}