Structure and minimal generating sets of Sylow 2-subgroups of alternating groups, properties of its commutator subgroup
| Autor: | Skuratovskii, Ruslan |
|---|---|
| Rok vydání: | 2017 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 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 |
| Databáze: | arXiv |
| Externí odkaz: |
|