Structure and minimal generating sets of Sylow 2-subgroups of alternating groups, properties of its commutator subgroup
Ruslan Skuratovskii
备注：
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
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.
Minimal generating set of Sylow 2-subgroups commutator subgroup of alternating group. Commutator width in Sylow $p$-subgroups of alternating, symmetric groups and in the wreath product of groups
Rank type conditions on commutators in finite groups
Università degli Studi di Modena e Reggio Emilia（摩德纳和雷焦艾米利亚大学）；University of Southern California（南加州大学）；University of Lincoln（林肯大学）；University of Brasilia（巴西利亚大学）
Sylow synchronization in finite groups: the good case
Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）；King’s College London（伦敦国王学院）；Heilbronn Institute for Mathematical Research（希尔布隆数学研究所）；University of Warwick（华威大学）
Local--global generation property of commutators in finite $π$-soluble groups
Università degli Studi di Modena e Reggio Emilia（摩德纳和雷焦艾米利亚大学）；University of Southern California（南加州大学）；University of Lincoln（林肯大学）；University of Brasilia（巴西利亚大学）
Ore revisited: an algorithmic investigation of the simple commutator promise problem
出于测试I. Anshel，M. Anshel和D. Goldfeld，（“公钥密码学的代数方法”，数学研究快报，第6卷，pp. 1-5，1999），我们研究了辫群B_n的简单交换子决策和承诺问题（SCDP/SCPP）的算法方法。我们把O. Ore，（"Some Remarks on Commutators“，Proceedings of the American Mathematical Society，Vol. 2，No. 2，pp.307-314，1951），研究了对称群的SCPP。我们的结果建立在H.采伊廷和我。Rivin，（``A Property of Alternating Groups''，arXiv：math.GR/0303036）.
