On certain minimal non-\(\mathfrak{Y}\)-groups for some classes \(\mathfrak{Y}\) (Q2813426)

Let \(\{\theta_n:n= 1,2,\dots\}\) be a sequence of words. Suppose \(G\) is an infinite locally finite group with trivial centre such that \(\theta_i(G)=G\) for all \(i\geq 1\). Assume that for every proper subgroup \(K\) of \(G\) there exists \(k\geq 1\) such that \(\theta_i(K)=\langle 1\rangle\) for every \(\geq k\).NEWLINENEWLINE Then, the authors prove that there exists \(t\geq 1\) such that for every proper normal subgroup \(N\) of \(G\) either \(\theta_t(N)=\langle 1\rangle\) or \(\theta_t(C_G(N))\langle 1\rangle\). The authors apply this result to various sequences of poly nilpotent words, of poly Engel words and of finite exponent words.