On the generalized dimension subgroup problems

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，我们记U∈μ_t(S)，如果对[U，A，A]=1的S的初等交换子群A，U=<u>.设G是有限群，S∈Syl_2(G)，V=&lt；u^q；u∈U，g∈G&gt；林给出了m的一个估计，使得当V是阿贝尔时，V∈μ_m(S)。设k是特征p&gt；0的域，G是有限群。群代数k[G]的Jacobson根的幂零指数t(G)是研究群代数的重要不变量之一。设G是有限p-可解群，P∈Syl_p(G)且p^m=|P|.Ninomiya对满足p^&lt；m-2&gt；&lt；t(G)&lt；p^&lt；m-1&gt；的群进行了分类。此外，当p=3时，他给出了t(G)&t(P)的例子群。这是对t(G)[小于或等于]t(P)总是成立的猜想的反例。在适当的条件下，Kawamoto证明了交换结合代数的单性与其导子李代数之间存在对偶关系。Knemitsu给出了无挠可消半群S的超半群在5上平坦的充要条件，设G是无挠可加群，S是G的子么半群，证明了如果S是具有线性序素理想的GCD-半群，则S是赋值半群。

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.