{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:16df3086-ee73-47de-8a9c-e0a17709c146", "labels": { "zh-CN": "拉格朗日定理", "en": "Lagrange’s theorem" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "d6629b2a00579007897ae8285ce9a10f7bb16e871645136e20db7838e482808d", "occurrences": [ { "concept_id": "urn:uuid:16df3086-ee73-47de-8a9c-e0a17709c146", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-02/05-normal-subgroups-and-factor-groups", "anchor": "first-e3b29e3a2bdd76ae5eeb-zh", "chapter": "2", "section": "2.5", "offset": 1546, "range": [ 6268, 6556 ], "source_revision": "fee321b0500c365016e4a1e97ab854a0117b68f310294f13cc853414839be4d8", "source_category": "translation", "review_status": "unreviewed", "role": "theorem", "role_basis": { "kind": "registered-source-environment", "registered_position": { "route": "zh/chapter-02/05-normal-subgroups-and-factor-groups", "lang": "zh-CN", "section": "2.5", "signal": "named-theorem", "label": "拉格朗日定理", "english": "Lagrange’s theorem", "offset": 1320 } }, "label": "> [!theorem] 定理 2.3 · 拉格朗日定理 (Lagrange’s theorem)", "context": { "range": [ 6268, 6556 ], "quote": "\n\n> [!theorem] 定理 2.3 · 拉格朗日定理 (Lagrange’s theorem)\n>\n> 如果$G$是一个阶数为$n$的有限群，那么每个子群$H$的阶数都是$n$的一个约数\n\n> [!proof]- 证明\n>\n> 每个陪集$gH$与$H$之间存在一一对应关系，因为如果$gh_1=gh_2$，那么$h_1=g^{-1}gh_2=h_2$。因此，每个陪集$gH$必须有完全相同的$|H|$个元素，由于陪集划分了群$G$，因此$n$是$|H|$的倍数。\n", "quote_sha256": "600a02512edd85685157d505c42e11bf0991e061b31d854660b8926d7674c80b", "environment": "theorem", "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-02/05-normal-subgroups-and-factor-groups", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-eb4fafa31520812410b645c8" }, { "concept_id": "urn:uuid:16df3086-ee73-47de-8a9c-e0a17709c146", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-02/05-normal-subgroups-and-factor-groups", "anchor": "first-e3b29e3a2bdd76ae5eeb-en", "chapter": "2", "section": "2.5", "offset": 2128, "range": [ 6341, 6872 ], "source_revision": "191e5ffe797661d4803bff76ea0bc213b1dcc7deb24b6a955143f84053aa53ee", "source_category": "original_transcription", "review_status": "unreviewed", "role": "theorem", "role_basis": { "kind": "registered-source-environment", "registered_position": { "route": "en/chapter-02/05-normal-subgroups-and-factor-groups", "lang": "en", "section": "2.5", "signal": "named-theorem", "label": "Lagrange’s theorem", "english": "Lagrange’s theorem", "offset": 2058 } }, "label": "> [!theorem] Theorem 2.3 · Lagrange’s theorem", "context": { "range": [ 6341, 6872 ], "quote": "\n\n> [!theorem] Theorem 2.3 · Lagrange’s theorem\n>\n> (Lagrange) If $G$ is afinite group oforder $n$, then the order ofevery subgroup $H$ is a divisor of $n$.\n\n> [!proof]- Proof\n>\n> Every coset $gH$ is in one-to-one correspondence with $H$, for if $g h _ { 1 } = g h _ { 2 }$ then $h _ { 1 } = g ^ { - 1 } g h _ { 2 } = h _ { 2 }$ . 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