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Moreover, we apply this theorem to prove that any group $G$ that acts without inversion on a tree $T$ that possesses a segment $\\Gamma$ for its quotient graph, such that, if the stabilizers of the vertex set $\\{\\,P,Q\\,\\}$ and edge $y$ of a lift of $\\, \\Gamma$ in $T$ are of the form $G_{P}\\!\\rtimes H$, $G_{Q}\\!\\rtimes H$ and $G_{y}\\! \\rtimes H$, then $G$ is isomorphic to the semidirect product "},"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":"1801.00057","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-12-29T23:09:02Z","cross_cats_sorted":[],"title_canon_sha256":"329fe84059b7cabfbd2f6b13f3de3f383f566b8b997d5f79a2ae6891d6f5dcb6","abstract_canon_sha256":"08d086bdbae74dce3e17eb26d3fa47a912eccc5dbe2d1850aea7055d812b9689"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:59.614432Z","signature_b64":"rEzGQQrVR7aqNB3or1JMCcnfYg5GNJcPLyMagCH0ErglKD7YdOaQWV5EHPOWsVJ4Ccjqq05SmcTmPvTpmfmyBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"21ac0dd3b6efe8e6e838b776ecb14a155d97ef381771a6c03e98f22da9314677","last_reissued_at":"2026-05-18T00:26:59.613781Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:59.613781Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Preservation of Trees by semidirect Products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Gabriel Zapata","submitted_at":"2017-12-29T23:09:02Z","abstract_excerpt":"We show that the semidirect product of a group $C$ by $A*_D B$ is isomorphic to the free product of $A\\rtimes C$ and $B\\rtimes C$ amalgamated at $D\\rtimes C$, where $A$, $B$ and $C$ are arbitrary groups. Moreover, we apply this theorem to prove that any group $G$ that acts without inversion on a tree $T$ that possesses a segment $\\Gamma$ for its quotient graph, such that, if the stabilizers of the vertex set $\\{\\,P,Q\\,\\}$ and edge $y$ of a lift of $\\, \\Gamma$ in $T$ are of the form $G_{P}\\!\\rtimes H$, $G_{Q}\\!\\rtimes H$ and $G_{y}\\! \\rtimes H$, then $G$ is isomorphic to the semidirect product "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.00057","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":"1801.00057","created_at":"2026-05-18T00:26:59.613883+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.00057v1","created_at":"2026-05-18T00:26:59.613883+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.00057","created_at":"2026-05-18T00:26:59.613883+00:00"},{"alias_kind":"pith_short_12","alias_value":"EGWA3U5W57UO","created_at":"2026-05-18T12:31:12.930513+00:00"},{"alias_kind":"pith_short_16","alias_value":"EGWA3U5W57UON2BY","created_at":"2026-05-18T12:31:12.930513+00:00"},{"alias_kind":"pith_short_8","alias_value":"EGWA3U5W","created_at":"2026-05-18T12:31:12.930513+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/EGWA3U5W57UON2BYW53OZMKKCV","json":"https://pith.science/pith/EGWA3U5W57UON2BYW53OZMKKCV.json","graph_json":"https://pith.science/api/pith-number/EGWA3U5W57UON2BYW53OZMKKCV/graph.json","events_json":"https://pith.science/api/pith-number/EGWA3U5W57UON2BYW53OZMKKCV/events.json","paper":"https://pith.science/paper/EGWA3U5W"},"agent_actions":{"view_html":"https://pith.science/pith/EGWA3U5W57UON2BYW53OZMKKCV","download_json":"https://pith.science/pith/EGWA3U5W57UON2BYW53OZMKKCV.json","view_paper":"https://pith.science/paper/EGWA3U5W","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.00057&json=true","fetch_graph":"https://pith.science/api/pith-number/EGWA3U5W57UON2BYW53OZMKKCV/graph.json","fetch_events":"https://pith.science/api/pith-number/EGWA3U5W57UON2BYW53OZMKKCV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EGWA3U5W57UON2BYW53OZMKKCV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EGWA3U5W57UON2BYW53OZMKKCV/action/storage_attestation","attest_author":"https://pith.science/pith/EGWA3U5W57UON2BYW53OZMKKCV/action/author_attestation","sign_citation":"https://pith.science/pith/EGWA3U5W57UON2BYW53OZMKKCV/action/citation_signature","submit_replication":"https://pith.science/pith/EGWA3U5W57UON2BYW53OZMKKCV/action/replication_record"}},"created_at":"2026-05-18T00:26:59.613883+00:00","updated_at":"2026-05-18T00:26:59.613883+00:00"}