{"paper":{"title":"On the Structure of Involutions and Symmetric Spaces of Quasi Dihedral Group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Bijan Davvaz, Imran, Zahid Raza","submitted_at":"2013-12-31T08:22:49Z","abstract_excerpt":"Let $G=QD_{8k}~$ be the quasi-dihedral group of order $8n$ and $\\theta$ be an automorphism of $QD_{8k}$ of finite order. The fixed-point set $H$ of $\\theta$ is defined as $H_{\\theta}=G^{\\theta}=\\{x\\in G \\mid \\theta(x)=x\\}$ and generalized symmetric space $Q$ of $\\theta$ given by $Q_{\\theta}=\\{g\\in G \\mid g=x\\theta(x)^{-1}~\\mbox{for some}~x\\in G\\}.$\n  The characteristics of the sets $H$ and $Q$ have been calculated. It is shown that for any $H$ and $Q,~~H.Q\\neq QD_{8k}.$ the $H$-orbits on $Q$ are obtained under different conditions. Moreover, the formula to find the order of $v$-th root of unit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.0112","kind":"arxiv","version":3},"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"}