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.

令t为非负整数，为有限的2组，而u为S的基本Abelian正常亚组。如果u = <u，我们写U∈μ_t（s）； | [u，a] | [少于或等于] 2^t>对于S s的任何基本Abelian子组a，[u，a，a] =1。令G为有限的组，s∈Syl_2（g）和V = <u^q; u∈U，g∈G>。 Hayashi给出了M的估计，以使V∈µ_M（S）每当V为Abelian时。令K为特征P> 0的领域，G为有限的组。该组代数K [g]的雅各布森激进分子的尼尔疗法指数t（g）是研究组代数的最重要的不变之一。令G为有限的P溶剂组，P∈Syl_p（G）和P^M = | P |。 Ninomiya对满足P^<m-2> <t（g）<p^<m-1>的所有组分类。此外，当p = 3时，他给了示例组，t（g）> t（p）。这是对t（g）[少于或相等] t（p）始终保持的猜想的反例。川本证明，在某些合适的条件下，换向性缔合代数的简单性与代数为代数之间存在双重性。 Knemitsu给出了一种必要且充分的条件，即无扭力的取消式半群的群体固定在5中。让G为无扭力的添加剂组，S为G。他证明，如果S是GCD-序列，则是GCD-序列，其具有线性命令的主要思想，然后S是一个价值的Semigroup。