{"paper":{"title":"On Covers of Dihedral 2-Groups by Powerful Subgroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Adam Gregory, Andrew Penland, Luke Guatelli, Risto Atanasov","submitted_at":"2019-01-10T20:24:55Z","abstract_excerpt":"A finite $p$-group $G$ is called \\textit{powerful} if either $p$ is odd and $[G,G]\\subseteq G^p$ or $p=2$ and $[G,G]\\subseteq G^4$. A {\\em{cover}} for a group is a collection of subgroups whose union is equal to the entire group. We will discuss covers of $p$-groups by powerful $p$-subgroups. The size of the smallest cover of a $p$-group by powerful $p$-subgroups is called the \\textit{powerful covering number}. In this paper we determine the powerful covering number of the dihedral 2-groups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.03372","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"}