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

Ruslan V. Skuratovskii

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2020), pp. 3-16

Résumé

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.