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Moreover, let $\\operatorname{Sym}(\\mathbb{F}_2^\\omega)$ denote the symmetric group acting on the elements of $\\mathbb{F}_2^\\omega$. It is shown that there are exactly four closed subgroups, $G$, such that $\\operatorname{GL}(\\omega, 2)\\leq G\\leq \\operatorname{Sym}(\\mathbb{F}_2^\\omega)$. As $\\mathbb{F}_2^\\omega$ is an $\\omega$-categorical (and homogeneous) structure, these groups correspond to the first order definable reducts of $"},"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":"1506.00220","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-05-31T12:28:51Z","cross_cats_sorted":[],"title_canon_sha256":"f0a6f085907924c464e5a97e87a27b4b5aadd8fd8c3883cce6b433222f0330f3","abstract_canon_sha256":"9ef05bb42cafad9c7f7b234de3f5bac0e027629f774014d33e24b0863a91a817"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:59:55.163239Z","signature_b64":"aLgjkU1bpzzIV/CFlp5oHB9LU4cC5eCwJhTNaHsuMz8ofZge3SbwQvwhVs9fEjGFYhY1G2SWZaMWYXKKDtLBCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b705a72ebaa6089d227edf7164f06c5cfa71749e00117355d95f285e0d51f57e","last_reissued_at":"2026-05-18T01:59:55.162678Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:59:55.162678Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Permutation groups containing infinite linear groups and reducts of infinite dimensional linear spaces over the two element field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Bertalan Bodor, Csaba Szab\\'o, Kende Kalina","submitted_at":"2015-05-31T12:28:51Z","abstract_excerpt":"Let $\\mathbb{F}_2^\\omega$ denote the countably infinite dimensional vector space over the two element field and $\\operatorname{GL}(\\omega, 2)$ its automorphism group. 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