{"title":"A note on finite group 1-cohomology via semi-direct products with applications to permutation modules","authors":"M. E. Harris","doi":"10.12988/ija.2021.91566","DOIUrl":null,"url":null,"abstract":"Section 17 of the important textbook [1] of M. Aschbacher studies Finite Group 1-Cohomology with a field coefficient ring via semi-direct products. This approach yields new structures and results to this basic subject. Here we assume that the coefficient ring is any commutative ring and we obtain all of the results of [1, Section 17] excluding Theorem 17.12. Via duality, this theorem extends the previous main result Theorem 17.11. In our final main results we assume that the coefficient ring is a discrete valuation ring, so that [1, Theorem 17.12] is a special case. Thus all of our results are applicable to Finite Group Modular Representation Theory. We conclude with applications to finite group permutation modules. Mathematics Subject Classification: 20J06","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Algebra and Computation","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.12988/ija.2021.91566","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Section 17 of the important textbook [1] of M. Aschbacher studies Finite Group 1-Cohomology with a field coefficient ring via semi-direct products. This approach yields new structures and results to this basic subject. Here we assume that the coefficient ring is any commutative ring and we obtain all of the results of [1, Section 17] excluding Theorem 17.12. Via duality, this theorem extends the previous main result Theorem 17.11. In our final main results we assume that the coefficient ring is a discrete valuation ring, so that [1, Theorem 17.12] is a special case. Thus all of our results are applicable to Finite Group Modular Representation Theory. We conclude with applications to finite group permutation modules. Mathematics Subject Classification: 20J06
M. Aschbacher的重要教科书[1]第17节通过半直积研究了具有场系数环的有限群1-上同。这种方法为这一基础学科提供了新的结构和结果。这里我们假设系数环是任意可交换环，我们得到了除定理17.12外的[1,Section 17]的所有结果。通过对偶性，这个定理扩展了前面的主要结果定理17.11。在我们最后的主要结果中，我们假设系数环是一个离散估值环，因此[1，定理17.12]是一个特例。因此，我们所有的结果都适用于有限群模表示理论。最后给出了有限群置换模的应用。数学学科分类:20J06
期刊介绍：
The International Journal of Algebra and Computation publishes high quality original research papers in combinatorial, algorithmic and computational aspects of algebra (including combinatorial and geometric group theory and semigroup theory, algorithmic aspects of universal algebra, computational and algorithmic commutative algebra, probabilistic models related to algebraic structures, random algebraic structures), and gives a preference to papers in the areas of mathematics represented by the editorial board.
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