Structure and minimal generating sets of Sylow 2-subgroups of alternating groups, properties of its commutator subgroup

TL;DR

This paper investigates the structure and minimal generating sets of Sylow 2-subgroups of alternating groups, proving minimality and describing their structure and commutator properties.

Contribution

It provides the first proof of minimality for generating sets of Sylow 2-subgroups of alternating groups and details their structural properties.

Findings

Established minimal generating sets for Sylow 2-subgroups

Described the structure of Sylow 2-subgroups of A_n

Analyzed the commutator width of Sylow 2-subgroups

Abstract

In this article the investigation of Sylows p-subgroups of ${{A}_{n}}$ and ${{S}_{n}}$, which was started in article of U. Dmitruk, V. Suschansky "Structure of 2-sylow subgroup of symmetric and alternating group" and article of R.~Skuratovskii "Corepresentation of a Sylow p-subgroup of a group S_n" \cite{Dm, Sk, Paw} is continued. Let $Syl_2{A_{2^k}}$ and $Syl_2{A_{n}}$ be Sylow 2-subgroups of corresponding alternating groups $A_{2^k}$ and $A_{n}$. We find a least generating set and a structure for such subgroups $Sy{{l}_{2}}{{A}_{{{2}^{k}}}}$ and $Syl_2{A_{n}}$ and commutator width of $Syl_2{A_{2^k}}$ \cite{Mur}. The authors of \cite{Dm, Paw} didn't proof minimality of finding by them system of generators for such Sylow 2-subgroups of $A_n$ and structure of it were founded only descriptively. The purpose of this paper is to research the structure of a Sylow 2-subgroups and to construct a minimal generating system for such subgroups. In other words, the problem is not simply in the proof of existence of a generating set with elements for the Sylow 2-subgroup of alternating group of degree $2^k $ and proof its minimality. The main result is the proof of minimality of this generating set of the above described subgroups and also the description of their structure.