{"paper":{"title":"Units of group rings, the Bogomolov multiplier, and the fake degree conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Andrei Jaikin-Zapirain, Javier Garcia-Rodriguez, Urban Jezernik","submitted_at":"2015-02-11T10:15:18Z","abstract_excerpt":"Let $\\pi$ be a finite $p$-group and $\\mathbb{F}_q$ a finite field with $q=p^n$ elements. Denote by $\\mathrm{I}_{\\mathbb{F}_q}$ the augmentation ideal of the group ring $\\mathbb{F}_q[\\pi]$. We have found a surprising relation between the abelianization of $1+\\mathrm{I}_{\\mathbb{F}_q}$, the Bogomolov multiplier $\\mathrm{B}_0(\\pi)$ of $\\pi$ and the number of conjugacy classes $\\mathrm{k}(\\pi)$ of $\\pi$: \\[ | (1+\\mathrm{I}_{\\mathbb{F}_q})_{\\mathrm{ab}} |=q^{\\mathrm{k}(\\pi)-1}|\\mathrm{B}_0(\\pi)|. \\] In particular, if $\\pi$ is a finite $p$-group with a non-trivial Bogomolov multiplier, then $1+\\math"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.03242","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"}