Rigid commutators and a normalizer chain

TL;DR

This paper introduces rigid commutators to analyze the normalizer chain in Sylow 2-subgroups of symmetric groups, linking the sequence of indices to Euler's partition theorem.

Contribution

It presents a novel concept of rigid commutators and connects the normalizer chain indices to integer partitions, extending understanding of Sylow 2-subgroups.

Findings

Sequence of indices corresponds to partial sums of partitions into distinct parts

Sequence relates to Euler's partition theorem

Provides a new combinatorial approach to group-theoretic structures

Abstract

The novel notion of rigid commutators is introduced to determine the sequence of the logarithms of the indices of a certain normalizer chain in the Sylow 2-subgroup of the symmetric group on 2^n letters. The terms of this sequence are proved to be those of the partial sums of the partitions of an integer into at least two distinct parts, that relates to a famous Euler's partition theorem.