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

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 $Sy{l}_2{A}_{2^k}$ and $Sy{l}_2{A}_n$ 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 $Sy{l}_2{A}_n$ and commutator width of $Sy{l}_2{A}_{2^k}$ \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.

It was presented by Skuratovlkii R. on 5 conferences: X International Algebraic Conference in Odessa dedicated to the 70th anniversary of Yu. A. Drozd. (2015), pp. 104., also Conference and PhD-Master Summer School on Graphs and Groups, Spectra and Symmetries. Novosibrsk 2016. arXiv admin note: substantial text overlap with arXiv:1612.02473, arXiv:1607.04855