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Further, we show that in the case where m is a prime-power the class of all groups G in which the w-values are n-Engel and v(G) has an ascending normal series whose quotients are either locally soluble or locally finite is a variety (Theorem B). We examine the question whether the latter result "},"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":"1503.07370","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-03-25T13:35:02Z","cross_cats_sorted":[],"title_canon_sha256":"e9e8b6809e6e1cb2df618b70ebf3d3646ca93807822ef8899807ddef2f77cc16","abstract_canon_sha256":"d38e735bb4b87e5ffb75e2cf8eb776cfcf3576d4155f1c66becc2525f9212840"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:20:21.272964Z","signature_b64":"RflZgpgQUIi3q/+s2yCN+pcxLu2/TiDjolMQQeZILLDG6CkvpXqRdfWU+Wvu3s+eRW8LJDn7me5wusxnNeTBAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"477e5e654c3afaee60eaadcbaadf47851a6ba52dae1afece7054fad57c607e41","last_reissued_at":"2026-05-18T02:20:21.272328Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:20:21.272328Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On varieties of groups satisfying an Engel type identity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"A. Tortora, M. Tota, P. Shumyatsky","submitted_at":"2015-03-25T13:35:02Z","abstract_excerpt":"Let m, n be positive integers, v a multilinear commutator word and w = v^m. Denote by v(G) and w(G) the verbal subgroups of a group G corresponding to v and w, respectively. We prove that the class of all groups G in which the w-values are n-Engel and w(G) is locally nilpotent is a variety (Theorem A). Further, we show that in the case where m is a prime-power the class of all groups G in which the w-values are n-Engel and v(G) has an ascending normal series whose quotients are either locally soluble or locally finite is a variety (Theorem B). We examine the question whether the latter result "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.07370","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":"1503.07370","created_at":"2026-05-18T02:20:21.272417+00:00"},{"alias_kind":"arxiv_version","alias_value":"1503.07370v1","created_at":"2026-05-18T02:20:21.272417+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.07370","created_at":"2026-05-18T02:20:21.272417+00:00"},{"alias_kind":"pith_short_12","alias_value":"I57F4ZKMHL5O","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_16","alias_value":"I57F4ZKMHL5O4YHK","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_8","alias_value":"I57F4ZKM","created_at":"2026-05-18T12:29:25.134429+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/I57F4ZKMHL5O4YHKVXF2VX2HQU","json":"https://pith.science/pith/I57F4ZKMHL5O4YHKVXF2VX2HQU.json","graph_json":"https://pith.science/api/pith-number/I57F4ZKMHL5O4YHKVXF2VX2HQU/graph.json","events_json":"https://pith.science/api/pith-number/I57F4ZKMHL5O4YHKVXF2VX2HQU/events.json","paper":"https://pith.science/paper/I57F4ZKM"},"agent_actions":{"view_html":"https://pith.science/pith/I57F4ZKMHL5O4YHKVXF2VX2HQU","download_json":"https://pith.science/pith/I57F4ZKMHL5O4YHKVXF2VX2HQU.json","view_paper":"https://pith.science/paper/I57F4ZKM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1503.07370&json=true","fetch_graph":"https://pith.science/api/pith-number/I57F4ZKMHL5O4YHKVXF2VX2HQU/graph.json","fetch_events":"https://pith.science/api/pith-number/I57F4ZKMHL5O4YHKVXF2VX2HQU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I57F4ZKMHL5O4YHKVXF2VX2HQU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I57F4ZKMHL5O4YHKVXF2VX2HQU/action/storage_attestation","attest_author":"https://pith.science/pith/I57F4ZKMHL5O4YHKVXF2VX2HQU/action/author_attestation","sign_citation":"https://pith.science/pith/I57F4ZKMHL5O4YHKVXF2VX2HQU/action/citation_signature","submit_replication":"https://pith.science/pith/I57F4ZKMHL5O4YHKVXF2VX2HQU/action/replication_record"}},"created_at":"2026-05-18T02:20:21.272417+00:00","updated_at":"2026-05-18T02:20:21.272417+00:00"}