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Moreover, we classify the solvable non-nilpotent Lie groups $G$ which are multiplication groups for $3$-dimensional simply connected topological loops $L$ and $\\hbox{dim} \\ G \\le 5$. These groups are direct products of proper connected Lie groups and have dimension $5$. We find also the inner mapping groups of $L$."},"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":"1507.01134","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-07-04T19:07:39Z","cross_cats_sorted":[],"title_canon_sha256":"2718f790ad0bf7d566535a50e54d6012cb8047e35f561ae7e1932b77d0b771ba","abstract_canon_sha256":"e06d6ceb18201c15024ca33e50b99c83a128e7591b45ae5063f89fc5b38a0222"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:37:18.044993Z","signature_b64":"kZgULKdKWSBa+4Bx0T19z1AHA4KwDDTRtfzdrAtTEffeYTC/AZMRquqSWPFbaliDBmHOugOdIOuK3nQ0Xw3GDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9ee48ca79f3f78b7dbe03d9065f4b59733d0814d17468d7cd02a8f96029ee211","last_reissued_at":"2026-05-18T01:37:18.044206Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:37:18.044206Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Three-dimensional topological loops with solvable multiplication groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"\\'Agota Figula","submitted_at":"2015-07-04T19:07:39Z","abstract_excerpt":"We prove that each $3$-dimensional connected topological loop $L$ having a solvable Lie group of dimension $\\le 5$ as the multiplication group of $L$ is centrally nilpotent of class $2$. Moreover, we classify the solvable non-nilpotent Lie groups $G$ which are multiplication groups for $3$-dimensional simply connected topological loops $L$ and $\\hbox{dim} \\ G \\le 5$. These groups are direct products of proper connected Lie groups and have dimension $5$. 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