{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:057330d9-9ff5-427f-a15c-73c814aa8a20", "labels": { "zh-CN": "凯莱定理", "en": "Cayley’s theorem" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "f3a1b9a051112fb1e1c3fdf046ce312ab626a6747fbb6f527a115507fa824676", "occurrences": [ { "concept_id": "urn:uuid:057330d9-9ff5-427f-a15c-73c814aa8a20", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-02/04-homomorphisms-and-isomorphisms", "anchor": "first-93da16ef5eea70dc3859-zh", "chapter": "2", "section": "2.4", "offset": 4566, "range": [ 9788, 10607 ], "source_revision": "98fad8e4f5b9d4869e8d0771f963e80d4413869dd8391f0598d49aecc884e4d4", "source_category": "translation", "review_status": "unreviewed", "role": "theorem", "role_basis": { "kind": "registered-source-environment", "registered_position": { "route": "zh/chapter-02/04-homomorphisms-and-isomorphisms", "lang": "zh-CN", "section": "2.4", "signal": "named-theorem", "label": "凯莱定理", "english": "Cayley’s theorem", "offset": 4113 } }, "label": "> [!theorem] 定理 2.2 · 凯莱定理 (Cayley’s theorem)", "context": { "range": [ 9788, 10607 ], "quote": "\n\n> [!theorem] 定理 2.2 · 凯莱定理 (Cayley’s theorem)\n>\n> 任意$n$阶有限群都同构于排列群\n\n> [!proof]- 证明\n>\n> 对于每一个$g\\in G$，定义映射$L_{g} : G\\to G$为左乘以$g$，\n>\n> $$\n> L_{g}(x)=gx\\quad\\mathrm{where}\\quad x\\in G.\n> $$\n>\n> 由于\n>\n> $$\n> gx=gx^{\\prime}\\Longrightarrow x=x^{\\prime}\\quad\\mathrm{and}\\quad x=L_{g}(g^{-1}x)\\quad\\mathrm{for~all~}x\\in G.\n> $$\n>\n> 因此，这个映射是一一对应和可逆的。$L_g$ 因此排列 $G=\\{g_1=e,g_2,\\ldots,g_n\\}$ 的元素，并且可以被认为是$S_n$中的一个成员。它有属性 $L_g\\circ L_h=L_{gh}$，因为\n>\n> $$\n> L_{g}\\circ L_{h}(x)=g(hx)=(gh)x=L_{gh}(x),\\quad\\forall x\\in G.\n> $$\n>\n> 因此，由 $\\varphi(g)=L_g$ 定义的映射 $\\varphi:G\\to S_n$ 是一个同态，\n>\n> $$\n> \\varphi(g)\\varphi(h)=L_{g}\\circ L_{h}=L_{gh}=\\varphi(gh).\n> $$\n>\n> 此外，$\\varphi$ 是一一对应的，因为如果 $\\varphi(g)=\\varphi(h)$ 那么 $g=L_g(e)=L_h(e)=h$ 。因此，$G$ 与 $\\varphi(G)\\subseteq S_{n}$ 的子群是同构的。\n", "quote_sha256": "db8f5beaa0d2a60585c0445857e1cec0bed898318390c5ed23516676766e0ba2", "environment": "theorem", "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-02/04-homomorphisms-and-isomorphisms", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-b78d76dec761e8644fb2233b" }, { "concept_id": "urn:uuid:057330d9-9ff5-427f-a15c-73c814aa8a20", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-02/04-homomorphisms-and-isomorphisms", "anchor": "first-93da16ef5eea70dc3859-en", "chapter": "2", "section": "2.4", "offset": 6496, "range": [ 11207, 12520 ], "source_revision": "10f4a2702b4445e5695e5ed84f130cd016b0d197740c1ecf9af4a30b2f586f2f", "source_category": "original_transcription", "review_status": "unreviewed", "role": "theorem", "role_basis": { "kind": "registered-source-environment", "registered_position": { "route": "en/chapter-02/04-homomorphisms-and-isomorphisms", "lang": "en", "section": "2.4", "signal": "named-theorem", "label": "Cayley’s theorem", "english": "Cayley’s theorem", "offset": 6355 } }, "label": "> [!theorem] Theorem 2.2 · Cayley’s theorem", "context": { "range": [ 11207, 12520 ], "quote": "\n\n> [!theorem] Theorem 2.2 · Cayley’s theorem\n>\n> (Cayley) Every finite group $G$ of order $n$ is isomorphic to a permutation group.\n\n> [!proof]- Proof\n>\n> For every $g \\in G$ define the map $L _ { g } : G \\to G$ to be left multiplication by $g ,$\n>\n> $$\n> L _ {g} (x) = g x \\quad \\text { where } \\quad x \\in G.\n> $$\n>\n> This map is one-to-one and onto since\n>\n> $$\n> g x = g x ^ {\\prime} \\Longrightarrow x = x ^ {\\prime} \\quad \\text { and } \\quad x = L _ {g} (g ^ {- 1} x) \\quad \\text { for all } x \\in G.\n> $$\n>\n> The map $L _ { g }$ therefore permutes the elements of $G = \\{ g _ { 1 } = e , g _ { 2 } , . . . , g _ { n } \\}$ and may be identified with a member of $S _ { n }$ . It has the property $L _ { g } \\circ L _ { h } = L _ { g h }$ , since\n>\n> $$\n> L _ {g} \\circ L _ {h} (x) = g (h x) = (g h) x = L _ {g h} (x), \\quad \\forall x \\in G.\n> $$\n>\n> Hence the map $\\varphi : G \\to S _ { n }$ defined by $\\varphi ( g ) = L _ { g }$ is a homomorphism,\n>\n> $$\n> \\varphi (g) \\varphi (h) = L _ {g} \\circ L _ {h} = L _ {g h} = \\varphi (g h).\n> $$\n>\n> Furthermore, $\\varphi$ is one-to-one, for if $\\varphi ( g ) = \\varphi ( h )$ then $g = L _ { g } ( e ) = L _ { h } ( e ) = h$ . Thus $G$ is isomorphic to the subgroup of $\\varphi ( G ) \\subseteq S _ { n }$ -\n", "quote_sha256": "5162297c92e09f0f6ebfc970168394fd1b5b84b68fee42b1e6bad5ec2e3047e3", "environment": "theorem", "excerpt_omitted": false }, "parallel_readings": [ { "route": "zh/chapter-02/04-homomorphisms-and-isomorphisms", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-b9c374baade0688f8e5beb1f" } ], "registered_positions": [ { "route": "en/chapter-02/04-homomorphisms-and-isomorphisms", "lang": "en", "section": "2.4", "signal": "named-theorem", "label": "Cayley’s theorem", "english": "Cayley’s theorem", "offset": 6355 }, { "route": "zh/chapter-02/04-homomorphisms-and-isomorphisms", "lang": "zh-CN", "section": "2.4", "signal": "named-theorem", "label": "凯莱定理", "english": "Cayley’s theorem", "offset": 4113 } ], "statements": [], "resolved_references": [ { "raw": "href=\"#theorem-2-2\"", "href": "#theorem-2-2", "range": [ 9831, 9850 ], "relation": "references", "target_route": "zh/chapter-02/04-homomorphisms-and-isomorphisms", "target_anchor": "theorem-2-2", "resolved": true, "reason": null, "source_revision": "98fad8e4f5b9d4869e8d0771f963e80d4413869dd8391f0598d49aecc884e4d4", "source_route": "zh/chapter-02/04-homomorphisms-and-isomorphisms", "occurrence_id": "occ-b78d76dec761e8644fb2233b" }, { "raw": "href=\"#theorem-2-2\"", "href": "#theorem-2-2", "range": [ 11250, 11269 ], "relation": "references", "target_route": "en/chapter-02/04-homomorphisms-and-isomorphisms", "target_anchor": "theorem-2-2", "resolved": true, "reason": null, "source_revision": "10f4a2702b4445e5695e5ed84f130cd016b0d197740c1ecf9af4a30b2f586f2f", "source_route": "en/chapter-02/04-homomorphisms-and-isomorphisms", "occurrence_id": "occ-b9c374baade0688f8e5beb1f" } ], "reading_order": [ "occ-b78d76dec761e8644fb2233b", "occ-b9c374baade0688f8e5beb1f" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 2, "located_occurrences": 2, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "2056a651d31653d928b8b941e2004e6f1bbc41bd7a12b2b0336bed40f50a122b", "optional_enrichments": [] }