A modular idealizer chain and unrefinability of partitions with repeated parts

TL;DR

This paper generalizes the concept of normalizer chains in Sylow p-subgroups using idealizers in Lie rings, linking partitions into parts with algebraic structures and conjecturing growth similarities with symmetric groups.

Contribution

It introduces a new framework for normalizer chains via idealizers in Lie rings, extending previous work beyond p=2 and connecting partition theory with algebraic normalizer structures.

Findings

Established a correspondence between idealizer and normalizer chains for m=2.

Proposed a conjecture that idealizer chains grow similarly to normalizer chains in symmetric groups.

Defined idealizers generated by partitions with limited parts in a Lie ring context.

Abstract

Recently Aragona et al. have introduced a chain of normalizers in a Sylow 2-subgroup of Sym(2^n), 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(p^n), 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 an evidence of this, we establish a correspondence between the two constructions in the case m=2.