**Authors:** Ruslan V. Skuratovskii

### Abstract

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 $(cw(G))$ \cite{Mur} of a wreath product of groups. A presentation in the form of wreath recursion \cite{Ne} 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 $Syl_2 S_{p^k}$.
%As a result a short proof of the fact that the commutator width of Sylow 2-subgroups of alternating group ${A_{{2^{k}}}}$, %permutation group $S_{{2^{k}}}$ and Sylow $p$-subgroups of $Syl_2 A_{p^k}$, where $k>2$, $Syl_2 S_{p^k}$ are equal to 1 is %obtained. 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.}

Kiev, 03056, Peremogy 37, KPI Igor Sikorsky Kiev

Polytechnic Institution, Ukraine.

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