On the generalized dimension subgroup problems

设 t 为非负整数，S 为有限 2 群，U 为 S 的初等阿贝尔正规子群。如果 U = <u ；则我们写 U ∈ μ_t(S) ； |[u，A]| 【小于或等于】 2^t> 对于 S 的任意基本交换子群 A，且 [U, A, A] = 1。设 G 为有限群，S ∈ Syl_2(G) 且 V = <u^q ； u Î U, g ÎG>。 Hayashi 给出了 m 的估计，使得每当 V 是交换矩阵时 V ∈ μ_m(S)。令 k 为特征 p>0 的域，G 为有限群。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。令 G 为有限 p 可解群，P ∈ Syl_p (G) 且 p^m =|P|。二宫对满足 p^<m-2> < t(G) < p^<m-1> 的所有组进行分类。此外，当 p = 3 时，他给出了 t(G) > t(P) 的示例组。这是t(G)【小于或等于】t(P)总是成立的猜想的反例。河本证明了在一定条件下交换结合代数的简单性与其推导李代数之间存在对偶性。 Knemitsu给出了一个充分必要条件，即无挠抵消半群S的超半群在5上平坦。设G是无挠可加群，S是G的子幺半群。他证明了如果S是具有线性有序素理想的GCD半群，则S是估值半群。

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.