Recently Aragona et al. have introduced a chain of normalizers in a Sylow 2-subgroup of Sym(2ⁿ), starting from an elementary abelian regular subgroup. They have shown that the indices of consecutive groups in the chain depend on the number of partitions into distinct parts and have given a description, by means of rigid commutators, of the first n − 2 terms in the chain. Moreover, they proved that the (n − 1)-th term of the chain is described by means of rigid commutators corresponding to unrefinable partitions into distinct parts. Although the mentioned chain can be defined in a Sylow p-subgroup of Sym(pⁿ), for p > 2 computing the chain of normalizers becomes a challenging task, in the absence of a suitable notion of rigid commutators. This problem is addressed here from an alternative point of view. We propose a more general framework for the normalizer chain, defining a chain of idealizers in a Lie ring over ℤ_m whose elements are represented by integer partitions. We show how the corresponding idealizers are generated by subsets of partitions into at most m − 1 parts and we conjecture that the idealizer chain grows as the normalizer chain in the symmetric group. As evidence of this, we establish a correspondence between the two constructions in the case m = 2.

最近阿拉戈纳等人。从基本阿贝尔正则子群开始，在 Sym(2 ⁿ )的 Sylow 2 子群中引入了标准化链。他们表明，链中连续组的索引取决于划分为不同部分的数量，并通过刚性交换器给出了链中前n -2 项的描述。此外，他们证明了链的第（n -1）项是通过刚性换向器来描述的，刚性换向器对应于不可细化的划分为不同的部分。尽管所提到的链可以在Sym( p ⁿ )的 Sylow p子群中定义，但对于p > 2 计算，在缺乏刚性换向器的合适概念的情况下，规范化器链成为一项具有挑战性的任务。这里从另一个角度来解决这个问题。_{我们为规范化器链提出了一个更通用的框架，在 ℤ m}上的李环中定义理想化器链，其元素由整数分区表示。我们展示了相应的理想化器是如何通过划分成最多m − 1 个部分的子集生成的，并且我们推测理想化器链随着对称组中的标准化器链而增长。作为这一点的证据，我们在m = 2的情况下建立了两种结构之间的对应关系。