Minimal Generating Set and Structure of the Wreath Product for Groups, Commutator of Wreath Product and the Fundamental Group of the Orbit Morse Function

Abstract

The size of a minimal generating set for the commutator subgroup of Sylow 2-subgroups of alternating group is found. The structure of commutator subgroup of Sylow 2-subgroups of the alternating group ${A_{{2^{k}}}}$ is investigated.It is shown that $(Syl_2 A_{2^k})^2 = Syl'_2 A_{2^k}, \, k>2$.It is proved that the commutator length of an arbitrary element of the iterated wreath product of cyclic groups $C_{p_i}, \, p_i\in \mathbb{N}$ equals to 1. The commutator width of direct limit of wreath product of cyclic groups is found. This paper presents upper bounds of the commutator width $(cw(G))$ of a wreath product of groups.A recursive presentation of Sylows $2$-subgroups $Syl_2(A_{{2^{k}}})$ of $A_{{2^{k}}}$ is introduced. As a result the short proof that the commutator width of Sylow 2-subgroups of alternating group ${A_{{2^{k}}}}$, permutation group ${S_{{2^{k}}}}$ and Sylow $p$-subgroups of $Syl_2 A_{p^k}$ ($Syl_2 S_{p^k}$) are equal to 1 is obtained.A 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.