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

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 $(Sy{l}_2{A}_{2^k}{)}^{\prime }$, $p>2$. 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 $(Sy{l}_2{A}_{2^k}{)}^{\prime }$, $(Sy{l}_2{S}_{2^k}{)}^{\prime }$ 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.

International conference in Ukraine, ATA12. Structure of commutant and centralizer, minimal generating sets of Sylow 2-subgroups $Sy{l}_2{A}_n$ of alternating and symmetric groups.(2017).https://www.imath.kiev.ua/~topology. arXiv admin note: text overlap with arXiv:1702.05784. Mal'tsev Meeting 2017