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Research articles

Commutator subgroup of Sylow 2-subgroups of alternating group and the commutator width in the wreath product

Ruslan V. Skuratovskii

Kiev, 03056, Peremogy 37, KPI Igor Sikorsky Kiev Polytechnic Institution, Ukraine

Аннотация: It is proved that the commutator length of an arbitrary element of the iterated wreath product of cyclic groups $C_{p_i}, ~ p_i\in \mathbb{N} $, is equal to $1$. The commutator width of direct limit of wreath product of cyclic groups is found. This paper gives upper bounds of the commutator width $(cw(G))$ [1] of a wreath product of groups. A presentation in the form of wreath recursion [6] of Sylow $2$-subgroups $Syl_2A_{{2^{k}}}$ of $A_{{2^k}}$ is introduced. As a corollary, we obtain a short proof of the result that the commutator width is equal to $1$ for Sylow $2$-subgroups of the alternating group ${A_{{2^{k}}}}$, where $k>2$, permutation group ${S_{{2^{k}}}}$ and for Sylow $p$-subgroups $Syl_2 A_{p^k}$ and $Syl_2 S_{p^k}$. The commutator width of permutational wreath product $B \wr C_n$ is investigated. An upper bound of the commutator width of permutational wreath product $B \wr C_n$ for an arbitrary group $B$ is found.

Ключевые слова и фразы: wreath product of groups, minimal generating set of the commutator subgroup of Sylow $2$-subgroups, commutator width of wreath product, commutator width of Sylow $p$-subgroups, commutator subgroup of alternating group.

MSC: 20B27, 20B22, 20F65, 20B07, 20E45

Поступила в редакцию: 01.06.2018
Исправленный вариант: 31.01.2020

Язык публикации: английский