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On the generalized dimension subgroup problems

关于广义维数子群问题

基本信息

批准号：
12640021
负责人：
HAYASHI Makoto
金额：
$ 1.54万
依托单位：
Aichi University of Education
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640021/
关键词：
radical nilpotency indices p-solvable group Witt type Lie algebra oversemigroup devided semigroup closure Witt type Lie algehra divided semigroup 整群環 Lisp Witt algebra Seminovmal semigroup Riemenn-Roch-Weil closure of a 2-group

项目摘要

Let t be a nonnegative integer, S be a finite 2-group and U be an elementary abelian normal subgroup of S. We write U ∈ μ_t(S) if U = <u ; |[u, A]| 【less than or equal】 2^t> for any elementary abelian subgroup A of S with [U, A, A] = 1. Let G be a finite group, S ∈ Syl_2(G) and V = <u^q ; u ∈ U, g ∈G>. Hayashi gave an estimation of m such that V ∈ μ_m(S) whenever V is abelian. Let k be a field of characteristic p>0 and G be a finite group. The nilpotency index t(G) of the Jacobson radical of the group algebra k[G] is one of the most important invariant to investigate the group algebra. Let G be a finite p-solvable group, P ∈ Syl_p (G) and p^m =|P|. Ninomiya classified all the groups satisfying p^<m-2> < t(G) < p^<m-1>. Moreover, when p = 3, he gave the example group with t(G) > t(P). This is a counterexample to the conjecture that t(G) 【less than or equal】 t(P) always holds. Kawamoto proved that there is a duality between the simplicity of commutative associative algebras and its derivation Lie algebras under some suitable conditions. Knemitsu gave a necessary and sufficient condition that an oversemigroup of a torsion-free cancellative semigroup S is flat over 5. Let G be a torsionfree additive group and S be a submonoid of G. He proved that If S is a GCD-semigroup which has prime ideals that are linearly ordered, then S is a valuation semigroup.

设t是非负整数，S是有限2-群，U是S的初等交换正规子群。记U ∈ μ_t（S），若U = <u ;|[u，A]|[小于或等于] 2 ^t>，对于S的任何初等阿贝尔子群A，其中[U，A，A] = 1。设G是有限群，S ∈ Syl_2（G），V = <u^q ; u ∈ U，g ∈G>. Hayashi给出了m的一个估计，使得V ∈ μ_m（S），只要V是交换的.设k是特征p>0的域，G是有限群.群代数k[G]的Jacobson根的幂零指数t（G）是研究群代数最重要的不变量之一。设G是有限p-可解群，P ∈ Syl_p（G）且p^m =| P|.二宫将所有满足p^<m-2>< t（G）< p^的组进行分类<m-1>。此外，当p = 3时，他给出了t（G）> t（P）的例子群。这是一个反例的猜想，t（G）[小于或等于] t（P）总是成立。Kawamoto证明了在适当的条件下，交换结合代数的单性与其导子李代数的单性之间存在对偶。Knemitsu给出了挠自由可消半群S的上半群在5上平坦的一个充要条件。设G是一个无挠加法群，S是G的一个子幺半群.他证明，如果S是一个GCD-半群，其中有素理想是线性有序的，那么S是一个评价半群。