{"paper":{"title":"A Generalization of the Hughes Subgroup","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Mario Sracic, Mark L. Lewis","submitted_at":"2018-09-20T10:48:13Z","abstract_excerpt":"Let $G$ be a finite group, $\\pi$ be a set of primes, and define $H_{\\pi}(G)$ to be the subgroup generated by all elements of $G$ which do not have prime order for every prime in $\\pi$. In this paper, we investigate some basic properties of $H_{\\pi}(G)$ and its relationship to the Hughes subgroup. We show that for most groups, only one of three possibilities occur: $H_{\\pi}(G) = 1$, $H_{\\pi}(G)=G$, or $H_{\\pi}(G) = H_{p}(G)$ for some prime $p \\in \\pi$. There is only one other possibility: $G$ is a Frobenius group whose Frobenius complement has prime order $p$, and whose Frobenius kernel, $F$, i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.07564","kind":"arxiv","version":2},"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"}