Commutators of Sylow subgroups of alternating and symmetric groups, commutator width in the wreath product of groups

This paper investigate bounds of the commutator width \cite {Mur} of a wreath product of two groups. The commutator width of direct limit of wreath product of cyclic groups are found. For given a permutational wreath product sequence of cyclic groups we investigate its commutator width and some properties of its commutator subgroup. It was proven that the commutator width of an arbitrary element of the wreath product of cyclic groups $C_{p_i}, \, p_i\in \mathbb{N} $ equals to 1. As a corollary, it is shown that the commutator width of Sylows $p$-subgroups of symmetric and alternating groups $p \geq 2$ are also equal to 1. The structure of commutator and second commutator of Sylows $2$-subgroups of symmetric and alternating groups were investigated. For an arbitraty group $B$ an upper bound of commutator width of $C_p \wr B$ was founded. For an arbitraty group $B$ commutator width of $C_p \wr B$ was founded. Also commutator width of Sylow 2-subgroups of alternating group ${A_{2^{k}}}$, permutation group ${S_{2^{k}}}$ are founded. The result of research are extended on subgroups $(Syl_2 {A_{2^{k}}})'$, $p\geq2$. The structure of commutator subgroup of Sylow 2-subgroups of symmetric and alternating groups is investigated. Portrait representation of $(Syl_2 {A_{2^{k}}})'$, $(Syl_2 {S_{2^{k}}})'$ was investigated.