{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:ZLWF35SWYV4YY6Q5G3LW3FSEHZ","short_pith_number":"pith:ZLWF35SW","canonical_record":{"source":{"id":"1801.00113","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-12-30T10:52:59Z","cross_cats_sorted":[],"title_canon_sha256":"57614e39a2aec2125e3d373de0a9cc57e16933592b8d58f3f172e8114f190ee4","abstract_canon_sha256":"c063cb90432922cb5935e5e4ba621055133a969c7e63b412bcdda9a6c81e0a4e"},"schema_version":"1.0"},"canonical_sha256":"caec5df656c5798c7a1d36d76d96443e69c7396ce4b3038ddd8aa93118b474cf","source":{"kind":"arxiv","id":"1801.00113","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.00113","created_at":"2026-05-18T00:26:59Z"},{"alias_kind":"arxiv_version","alias_value":"1801.00113v1","created_at":"2026-05-18T00:26:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.00113","created_at":"2026-05-18T00:26:59Z"},{"alias_kind":"pith_short_12","alias_value":"ZLWF35SWYV4Y","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_16","alias_value":"ZLWF35SWYV4YY6Q5","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_8","alias_value":"ZLWF35SW","created_at":"2026-05-18T12:31:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:ZLWF35SWYV4YY6Q5G3LW3FSEHZ","target":"record","payload":{"canonical_record":{"source":{"id":"1801.00113","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-12-30T10:52:59Z","cross_cats_sorted":[],"title_canon_sha256":"57614e39a2aec2125e3d373de0a9cc57e16933592b8d58f3f172e8114f190ee4","abstract_canon_sha256":"c063cb90432922cb5935e5e4ba621055133a969c7e63b412bcdda9a6c81e0a4e"},"schema_version":"1.0"},"canonical_sha256":"caec5df656c5798c7a1d36d76d96443e69c7396ce4b3038ddd8aa93118b474cf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:59.379510Z","signature_b64":"Fpsq/vOMNH/N+Aou+vQHce3KsF1qJKU+6adTkRXxfsYEZsdYe6mQFYJjlBqmYfQgyWRWuixWHoQa1+L5za1RAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"caec5df656c5798c7a1d36d76d96443e69c7396ce4b3038ddd8aa93118b474cf","last_reissued_at":"2026-05-18T00:26:59.378738Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:59.378738Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1801.00113","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:26:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SS5PZJa6/A2S+kT22JCqGrmWWyXG+VYaRVMLV15u5cl0cqfeU2bghxroi5E+vqBS3GPuizlacqubLlILv204DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T22:05:30.670934Z"},"content_sha256":"629a1efcaec2ff7737710a78f3bd0f002e137f2f8c60f64b7d07d6a9b32642bc","schema_version":"1.0","event_id":"sha256:629a1efcaec2ff7737710a78f3bd0f002e137f2f8c60f64b7d07d6a9b32642bc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:ZLWF35SWYV4YY6Q5G3LW3FSEHZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A generalization of Neumann's Question","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"A. Ahmadkhah, M. Zarrin, S. Marzang","submitted_at":"2017-12-30T10:52:59Z","abstract_excerpt":"Let $G$ be a group, $m\\geq2$ and $n\\geq1$. We say that $G$ is an $\\mathcal{T}(m,n)$-group if for every $m$ subsets $X_1, X_2, \\dots, X_m$ of $G$ of cardinality $n$, there exists $i\\neq j$ and $x_i \\in X_i, x_j \\in X_j$ such that $x_ix_j=x_jx_i$. In this paper, we give some examples of finite and infinite non-abelian $\\mathcal{T}(m,n)$-groups and we discuss finiteness and commutativity of such groups. We also show solvability length of a solvable $\\mathcal{T}(m,n)$-group is bounded in terms of $m$ and $n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.00113","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:26:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O/prkz7KuBd9A2ZDj3JHhfwHa8TavMhsJ34pWBr+y9tndrsMx2TvJItKD5OSp6UUctycTdqhAO8ce1k9oGGMDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T22:05:30.671267Z"},"content_sha256":"9c81a66ae73ec653135437345a35cacc5aed51e1eda737140ed1681536b6fc1d","schema_version":"1.0","event_id":"sha256:9c81a66ae73ec653135437345a35cacc5aed51e1eda737140ed1681536b6fc1d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZLWF35SWYV4YY6Q5G3LW3FSEHZ/bundle.json","state_url":"https://pith.science/pith/ZLWF35SWYV4YY6Q5G3LW3FSEHZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZLWF35SWYV4YY6Q5G3LW3FSEHZ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-23T22:05:30Z","links":{"resolver":"https://pith.science/pith/ZLWF35SWYV4YY6Q5G3LW3FSEHZ","bundle":"https://pith.science/pith/ZLWF35SWYV4YY6Q5G3LW3FSEHZ/bundle.json","state":"https://pith.science/pith/ZLWF35SWYV4YY6Q5G3LW3FSEHZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZLWF35SWYV4YY6Q5G3LW3FSEHZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:ZLWF35SWYV4YY6Q5G3LW3FSEHZ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"c063cb90432922cb5935e5e4ba621055133a969c7e63b412bcdda9a6c81e0a4e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-12-30T10:52:59Z","title_canon_sha256":"57614e39a2aec2125e3d373de0a9cc57e16933592b8d58f3f172e8114f190ee4"},"schema_version":"1.0","source":{"id":"1801.00113","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1801.00113","created_at":"2026-05-18T00:26:59Z"},{"alias_kind":"arxiv_version","alias_value":"1801.00113v1","created_at":"2026-05-18T00:26:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.00113","created_at":"2026-05-18T00:26:59Z"},{"alias_kind":"pith_short_12","alias_value":"ZLWF35SWYV4Y","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_16","alias_value":"ZLWF35SWYV4YY6Q5","created_at":"2026-05-18T12:31:59Z"},{"alias_kind":"pith_short_8","alias_value":"ZLWF35SW","created_at":"2026-05-18T12:31:59Z"}],"graph_snapshots":[{"event_id":"sha256:9c81a66ae73ec653135437345a35cacc5aed51e1eda737140ed1681536b6fc1d","target":"graph","created_at":"2026-05-18T00:26:59Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $G$ be a group, $m\\geq2$ and $n\\geq1$. We say that $G$ is an $\\mathcal{T}(m,n)$-group if for every $m$ subsets $X_1, X_2, \\dots, X_m$ of $G$ of cardinality $n$, there exists $i\\neq j$ and $x_i \\in X_i, x_j \\in X_j$ such that $x_ix_j=x_jx_i$. In this paper, we give some examples of finite and infinite non-abelian $\\mathcal{T}(m,n)$-groups and we discuss finiteness and commutativity of such groups. We also show solvability length of a solvable $\\mathcal{T}(m,n)$-group is bounded in terms of $m$ and $n$.","authors_text":"A. Ahmadkhah, M. Zarrin, S. Marzang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-12-30T10:52:59Z","title":"A generalization of Neumann's Question"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.00113","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:629a1efcaec2ff7737710a78f3bd0f002e137f2f8c60f64b7d07d6a9b32642bc","target":"record","created_at":"2026-05-18T00:26:59Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"c063cb90432922cb5935e5e4ba621055133a969c7e63b412bcdda9a6c81e0a4e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-12-30T10:52:59Z","title_canon_sha256":"57614e39a2aec2125e3d373de0a9cc57e16933592b8d58f3f172e8114f190ee4"},"schema_version":"1.0","source":{"id":"1801.00113","kind":"arxiv","version":1}},"canonical_sha256":"caec5df656c5798c7a1d36d76d96443e69c7396ce4b3038ddd8aa93118b474cf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"caec5df656c5798c7a1d36d76d96443e69c7396ce4b3038ddd8aa93118b474cf","first_computed_at":"2026-05-18T00:26:59.378738Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:26:59.378738Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Fpsq/vOMNH/N+Aou+vQHce3KsF1qJKU+6adTkRXxfsYEZsdYe6mQFYJjlBqmYfQgyWRWuixWHoQa1+L5za1RAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:26:59.379510Z","signed_message":"canonical_sha256_bytes"},"source_id":"1801.00113","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:629a1efcaec2ff7737710a78f3bd0f002e137f2f8c60f64b7d07d6a9b32642bc","sha256:9c81a66ae73ec653135437345a35cacc5aed51e1eda737140ed1681536b6fc1d"],"state_sha256":"b7f0cb9822ca5e1e52e4b524665fd74cdbf0b08518324beccef0eb1beccf7e25"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K0JgPO2tPrCfkCwfluEhaaTyiBm2XXjTRPAPGT1YlMtepva9soQ1QuLgTlmlENeTEyN2SLl9jkhh216WYJmhDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T22:05:30.673167Z","bundle_sha256":"6a8ebd8b43bd4c942a8ab54618409a2c65046a390254702150853cc7de5cbaa2"}}