7 problems

Azevedo–Shumyatsky conjecture. The word

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…

Let $w$ be a group-word and $G$ a profinite group. Suppose that the set of $w$-values in $G$ is countable. Identity conjecture. Then $G$ satisfies a nontrivial group identity. The…

Let $w$ be a group-word and $G$ a profinite group. Suppose that the set of $w$-values in $G$ is countable. Profinite conciseness conjecture. Then the verbal subgroup $w(G)$ is fini…

Let $\Sigma$ be a property of profinite groups such that every finite group is a $\Sigma$-group, the class of $\Sigma$-groups is closed under taking subgroups, quotients and extens…

Let $w$ be a group-word and let $n$ be a positive integer. For a group $G$, write $w(G)$ for the verbal subgroup generated by all values of $w$ in $G$. An element is $n$-Engel if,…