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The size of minimal generating set for commutator of Sylow 2-subgroup ofalternating group was found. Given a permutational wreath product of finitecyclic groups sequence we prove that the commutator width of such groups is 1and we research some properties of its commutator subgroup. It was shown that$(Syl_2 A_{2^k})^2 = Syl'_2 (A_{2^k}), \, k>2$.  A new approach to presentation of Sylow 2-subgroups of alternating group${A_{{2^{k}}}}$ was applied. As a result the short proof that the commutatorwidth of Sylow 2-subgroups of alternating group ${A_{{2^{k}}}}$, permutationgroup ${S_{{2^{k}}}}$ and Sylow $p$-subgroups of $Syl_2 A_{p^k}$ ($Syl_2S_{p^k}$) are equal to 1 was obtained. Commutator width of permutational wreathproduct $B \wr C_n$ were investigated. It was proven that the commutator lengthof an arbitrary element of commutator of the wreath product of cyclic groups$C_{p_i}, \, p_i\in \mathbb{N} $ equals to 1. The commutator width of directlimit of wreath product of cyclic groups are found. As a corollary, it wasshown that the commutator width of Sylows $p$-subgroups $Syl_2(S_{{p^{k}}})$ ofsymmetric $S_{{p^{k}}}$ and alternating groups $A_{{p^{k}}}$ $p \geq 2$ arealso equal to 1. A recursive presentation of Sylows $2$-subgroups$Syl_2(A_{{2^{k}}})$ of $A_{{2^{k}}}$ was introduced. The structure of Sylows$2$-subgroups commutator of symmetric and alternating groups were investigated.For an arbitrary group $B$ an upper bound of commutator width of $C_p \wr B$was founded.

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