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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1401.0112","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-12-31T08:22:49Z","cross_cats_sorted":[],"title_canon_sha256":"ba8be440b09f9105b1a373bb2a74902355d729cf5225272a8dcf0a2212f1a481","abstract_canon_sha256":"fccc1dc40abb68b06020b6c6344937dd98813302f2ba84a43ead1fb3ca8fc96d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:41:02.575414Z","signature_b64":"9aYXYX8YWmsWuck97zC/vNqdX72fD0JbDT6QgpQBKPRZ1LwyI4cKOXg9YqJsBCJRW3XuZn/8IeS6WSao1ub4BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"843b2ae0943b646d7560211eeabbba2522a0291dc13a5f850bdcbcf79e97f74f","last_reissued_at":"2026-05-18T00:41:02.574818Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:41:02.574818Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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