{"title":"有限阿贝尔群的组合不变量","authors":"W.R. Emerson","doi":"10.1016/S0021-9800(70)80056-4","DOIUrl":null,"url":null,"abstract":"<div><p>If <em>G</em> is any finite Abelian group define<span><span><span><math><mrow><mi>γ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mstyle><munder><mo>∑</mo><mi>i</mi></munder><mrow><mo>(</mo><msub><mi>e</mi><mi>i</mi></msub><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mstyle></mrow></math></span></span></span></p><p>where <em>e<sub>i</sub></em> are the canonic invariants of <em>G</em> (<em>e</em><sub>i</sub>+1|<em>e</em><sub>i</sub> for all <em>i</em>). The primary result is the “super-additivity” of <em>γ</em>, i.e.,<span><span><span><math><mrow><mi>γ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>⩾</mo><mi>γ</mi><mrow><mo>(</mo><mrow><mrow><mi>G</mi><mo>/</mo><mi>H</mi></mrow></mrow><mo>)</mo></mrow><mo>+</mo><mi>γ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>for</mo><mo>all</mo><mo>subgroups</mo><mo>H</mo><mo>of</mo><mo>G</mo><mo>.</mo><mrow><mo>(</mo><mo>∗</mo><mo>)</mo></mrow></mrow></math></span></span></span></p><p>In the process of establishing (*) the structure of Abelian groups is studied in great detail and a technique is developed for proving inequalities analogous to (*) for other invariants than γ.</p><p>We then apply (*) to obtain various further results of a combinatorial nature.</p></div>","PeriodicalId":100765,"journal":{"name":"Journal of Combinatorial Theory","volume":"9 1","pages":"Pages 72-92"},"PeriodicalIF":0.0000,"publicationDate":"1970-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80056-4","citationCount":"1","resultStr":"{\"title\":\"A combinatorial invariant for finite Abelian groups with various applications\",\"authors\":\"W.R. Emerson\",\"doi\":\"10.1016/S0021-9800(70)80056-4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>If <em>G</em> is any finite Abelian group define<span><span><span><math><mrow><mi>γ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mstyle><munder><mo>∑</mo><mi>i</mi></munder><mrow><mo>(</mo><msub><mi>e</mi><mi>i</mi></msub><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mstyle></mrow></math></span></span></span></p><p>where <em>e<sub>i</sub></em> are the canonic invariants of <em>G</em> (<em>e</em><sub>i</sub>+1|<em>e</em><sub>i</sub> for all <em>i</em>). The primary result is the “super-additivity” of <em>γ</em>, i.e.,<span><span><span><math><mrow><mi>γ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>⩾</mo><mi>γ</mi><mrow><mo>(</mo><mrow><mrow><mi>G</mi><mo>/</mo><mi>H</mi></mrow></mrow><mo>)</mo></mrow><mo>+</mo><mi>γ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>for</mo><mo>all</mo><mo>subgroups</mo><mo>H</mo><mo>of</mo><mo>G</mo><mo>.</mo><mrow><mo>(</mo><mo>∗</mo><mo>)</mo></mrow></mrow></math></span></span></span></p><p>In the process of establishing (*) the structure of Abelian groups is studied in great detail and a technique is developed for proving inequalities analogous to (*) for other invariants than γ.</p><p>We then apply (*) to obtain various further results of a combinatorial nature.</p></div>\",\"PeriodicalId\":100765,\"journal\":{\"name\":\"Journal of Combinatorial Theory\",\"volume\":\"9 1\",\"pages\":\"Pages 72-92\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1970-07-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/S0021-9800(70)80056-4\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0021980070800564\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021980070800564","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A combinatorial invariant for finite Abelian groups with various applications
If G is any finite Abelian group define
where ei are the canonic invariants of G (ei+1|ei for all i). The primary result is the “super-additivity” of γ, i.e.,
In the process of establishing (*) the structure of Abelian groups is studied in great detail and a technique is developed for proving inequalities analogous to (*) for other invariants than γ.
We then apply (*) to obtain various further results of a combinatorial nature.
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