A Chain of Normalizers in the Sylow $2$-subgroups of the symmetric group on $2^n$ letters

TL;DR

This paper investigates a chain of normalizers within Sylow 2-subgroups of the symmetric group on 2^n letters, revealing a conjectured pattern in their indices related to partition numbers, with implications for symmetric cryptography.

Contribution

It introduces a novel chain of subgroups based on normalizers in symmetric groups and conjectures a pattern in their indices linked to partition functions.

Findings

Partial results support the conjecture for large n.

Indices of normalizers relate to partial sums of partition numbers.

Computational experiments suggest a stable pattern independent of n for large n.

Abstract

On the basis of an initial interest in symmetric cryptography, in the present work we study a chain of subgroups. Starting from a Sylow $2$-subgroup of AGL(2,n), each term of the chain is defined as the normalizer of the previous one in the symmetric group on $2^n$ letters. Partial results and computational experiments lead us to conjecture that, for large values of $n$, the index of a normalizer in the consecutive one does not depend on $n$. Indeed, there is a strong evidence that the sequence of the logarithms of such indices is the one of the partial sums of the numbers of partitions into at least two distinct parts.