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Then the dihedral-like automorphic loop $\\mathrm{Dih}(m,G,\\alpha)$ is defined on $\\mathbb Z_m\\times G$ by $(i,u)(j,v)=(i+j, ((-1)^{j}u+v)\\alpha^{ij})$. We prove that two finite dihedral-like automorphic loops $\\mathrm{Dih}(m,G,\\alpha)$, $\\mathrm{Dih}(\\overline{m},\\overline{G},\\overline{\\alpha})$ are isomorphic if and only if $m=\\ove"},"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":"1712.06516","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-12-18T16:51:28Z","cross_cats_sorted":[],"title_canon_sha256":"204fa3a435a382cefe9e753d56ddecc2473667906262116e8ca08fddf0de0c98","abstract_canon_sha256":"ff90ef978e02152ae87870c1382ab396d58615f06a3f7a25984a8964bfa8a60b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:27:47.646067Z","signature_b64":"pEeXvXQ2HxYTbKgE+d99jfPRNVH1dHphbbH3Gu1UrLHHAU01svplCJLFxinl8E5EI2DVeOIYvedtIjnzTf7yAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ffa90eda945e32dcef6087f697e18e54ddb8cb627a2b226439186aa464da5426","last_reissued_at":"2026-05-18T00:27:47.645611Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:27:47.645611Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Automorphisms of dihedral-like automorphic loops","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Mouna Aboras, Petr Vojt\\v{e}chovsk\\'y","submitted_at":"2017-12-18T16:51:28Z","abstract_excerpt":"Automorphic loops are loops in which all inner mappings are automorphisms. A large class of automorphic loops is obtained as follows: Let $m$ be a positive even integer, $G$ an abelian group, and $\\alpha$ an automorphism of $G$ that satisfies $\\alpha^2=1$ if $m>2$. Then the dihedral-like automorphic loop $\\mathrm{Dih}(m,G,\\alpha)$ is defined on $\\mathbb Z_m\\times G$ by $(i,u)(j,v)=(i+j, ((-1)^{j}u+v)\\alpha^{ij})$. We prove that two finite dihedral-like automorphic loops $\\mathrm{Dih}(m,G,\\alpha)$, $\\mathrm{Dih}(\\overline{m},\\overline{G},\\overline{\\alpha})$ are isomorphic if and only if $m=\\ove"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.06516","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1712.06516","created_at":"2026-05-18T00:27:47.645682+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.06516v1","created_at":"2026-05-18T00:27:47.645682+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.06516","created_at":"2026-05-18T00:27:47.645682+00:00"},{"alias_kind":"pith_short_12","alias_value":"76UQ5WUULYZN","created_at":"2026-05-18T12:31:03.183658+00:00"},{"alias_kind":"pith_short_16","alias_value":"76UQ5WUULYZNZ33A","created_at":"2026-05-18T12:31:03.183658+00:00"},{"alias_kind":"pith_short_8","alias_value":"76UQ5WUU","created_at":"2026-05-18T12:31:03.183658+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/76UQ5WUULYZNZ33AQ73JPYMOKT","json":"https://pith.science/pith/76UQ5WUULYZNZ33AQ73JPYMOKT.json","graph_json":"https://pith.science/api/pith-number/76UQ5WUULYZNZ33AQ73JPYMOKT/graph.json","events_json":"https://pith.science/api/pith-number/76UQ5WUULYZNZ33AQ73JPYMOKT/events.json","paper":"https://pith.science/paper/76UQ5WUU"},"agent_actions":{"view_html":"https://pith.science/pith/76UQ5WUULYZNZ33AQ73JPYMOKT","download_json":"https://pith.science/pith/76UQ5WUULYZNZ33AQ73JPYMOKT.json","view_paper":"https://pith.science/paper/76UQ5WUU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.06516&json=true","fetch_graph":"https://pith.science/api/pith-number/76UQ5WUULYZNZ33AQ73JPYMOKT/graph.json","fetch_events":"https://pith.science/api/pith-number/76UQ5WUULYZNZ33AQ73JPYMOKT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/76UQ5WUULYZNZ33AQ73JPYMOKT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/76UQ5WUULYZNZ33AQ73JPYMOKT/action/storage_attestation","attest_author":"https://pith.science/pith/76UQ5WUULYZNZ33AQ73JPYMOKT/action/author_attestation","sign_citation":"https://pith.science/pith/76UQ5WUULYZNZ33AQ73JPYMOKT/action/citation_signature","submit_replication":"https://pith.science/pith/76UQ5WUULYZNZ33AQ73JPYMOKT/action/replication_record"}},"created_at":"2026-05-18T00:27:47.645682+00:00","updated_at":"2026-05-18T00:27:47.645682+00:00"}