2 problems

Let $w=w(x_1,\dots,x_k)$ be a group-word, and let $G_w$ denote the set of all values of $w$ in a group $G$. The verbal subgroup $w(G)$ is the subgroup generated by $G_w$, and $w$ i…

Let $w$ be a multilinear commutator word and let $G$ be a finite group in which $|ab|=|a||b|$ whenever the $w$-values $a,b$ have coprime orders. Shumyatsky's conjecture. Then…