On Palindromic Widths of Nilpotent and Wreathe Products

We prove that the nilpotent product of a set of groups $A_1,\dots ,A_s$ has finite palindromic width if and only if the palindromic widths of $A_{i}, i=1,\dots, s,$arefinite\mathrm{.}Wegiveanewproofthatthecommutatorwidthof$F_n \wr K$isinfinite,where$F_n$isafreegroupofrank$n \geq 2$and$K$ a finite group. This result, combining with a result of Fink \cite{f1} gives examples of groups with infinite commutator width but finite palindromic width with respect to some generating set.