The commutator and centralizer description of Sylow 2-subgroups of alternating and symmetric groups

TL;DR

This paper investigates the structure and properties of Sylow 2-subgroups of symmetric and alternating groups, focusing on their commutator width, subgroup construction, and minimal generating sets, extending known results to broader classes.

Contribution

It provides a detailed construction of the commutator subgroups of Sylow 2-subgroups and extends the understanding of their properties to subgroups with p>2, including minimal generating sets.

Findings

The commutator width of Sylow 2-subgroups is explicitly characterized.

Construction methods for commutator subgroups are developed.

The minimal generic sets of Sylow 2-subgroups are identified.

Abstract

Given a permutational wreath product sequence of cyclic groups of prime order we research a commutator width of such groups and some properties of its commutator subgroup. Commutator width of Sylow 2-subgroups of alternating group $A_{2^{k}}$, permutation group $S_{2^k}$ and $C_p \wr B$ were founded. The result of research was extended on subgroups $(Syl_2 {A_{2^k}})'$, $p>2$. The paper presents a construction of commutator subgroup of Sylow 2-subgroups of symmetric and alternating groups. Also minimal generic sets of Sylow 2-subgroups of $A_{2^k}$ were founded. Elements presentation of $(Syl_2 {A_{2^k}})'$, $(Syl_2 {S_{2^k}})'$ was investigated. We prove that the commutator width \cite {Mur} of an arbitrary element of a discrete wreath product of cyclic groups $C_p$ is 1. Key words: wreath product of group, commutator width of $p$-Sylow subgroups, commutator subgroup, centralizer subgroup, semidirect product.