{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:XSS7R32SRA6FUZMTAM4ZBCKUT4","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":"ec0ad3b1828e2f2374bbd629f191947cae0a326708d2402a3ffb0765fbb28e06","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-07-21T18:21:28Z","title_canon_sha256":"9d3f248f01b23fc0d81245ff920124d9b3baabf5568748d6ee0605ff4ee7d640"},"schema_version":"1.0","source":{"id":"1607.06418","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.06418","created_at":"2026-05-18T01:10:41Z"},{"alias_kind":"arxiv_version","alias_value":"1607.06418v1","created_at":"2026-05-18T01:10:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.06418","created_at":"2026-05-18T01:10:41Z"},{"alias_kind":"pith_short_12","alias_value":"XSS7R32SRA6F","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XSS7R32SRA6FUZMT","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XSS7R32S","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:89fd0491c5b593871b99a4f49c3b7960502b5d29e087b558a280a577adb38148","target":"graph","created_at":"2026-05-18T01:10:41Z","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":"A set $B$ is a basis for a vector space $V$ if every element of $V$ can be uniquely written as a linear combination of the elements of $B$. There is a similar definition of a basis for a finite group. We show that certain semidirect products of finite groups---including all semidirect products of finite abelian groups---have bases; any group of order $m$ or $2m$ for odd, cube-free $m$ has a basis; and the quaternions do not have a basis.","authors_text":"Bret Benesh, Jason Lutz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-07-21T18:21:28Z","title":"Group bases for some solvable groups and semidirect products"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.06418","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":"sha256:7ed0b9ae7dc63d159984e246639a76ba10dec690f2bc351d26c3fc4262258e5f","target":"record","created_at":"2026-05-18T01:10:41Z","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":"ec0ad3b1828e2f2374bbd629f191947cae0a326708d2402a3ffb0765fbb28e06","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-07-21T18:21:28Z","title_canon_sha256":"9d3f248f01b23fc0d81245ff920124d9b3baabf5568748d6ee0605ff4ee7d640"},"schema_version":"1.0","source":{"id":"1607.06418","kind":"arxiv","version":1}},"canonical_sha256":"bca5f8ef52883c5a659303399089549f14e8c7b3739c7ec3e5e2f6eebdb59385","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bca5f8ef52883c5a659303399089549f14e8c7b3739c7ec3e5e2f6eebdb59385","first_computed_at":"2026-05-18T01:10:41.877335Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:10:41.877335Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wrV6mWhnctLI8iMxXHesRBLvdp0VazLkDItp+Ffpp64yAkVs57gPjyG/r03hWBR+pn87PsGDyOsI8jQgodW/CA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:10:41.877753Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.06418","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7ed0b9ae7dc63d159984e246639a76ba10dec690f2bc351d26c3fc4262258e5f","sha256:89fd0491c5b593871b99a4f49c3b7960502b5d29e087b558a280a577adb38148"],"state_sha256":"e692fa78b99766ef266d0ece64c481a5e96da8ee02ba84466102f563dd842593"}