4 problems

Hartley's conjecture. If the unit group $U(FG)$ satisfies a group identity, then $FG$ satisfies a polynomial identity.

For a positive integer $d$, let $\operatorname{F}(X_1,\ldots,X_d)$ be the free group on $X_1,\ldots,X_d$, and let $w\in\operatorname{F}(X_1,\ldots,X_d)$ be a nontrivial word. For a…

Let $D$ be a division ring with center $F$, and let $\operatorname{GL}_n(D)$ be the general linear group of degree $n$ over $D$. If $n\geq 2$, assume that $D$ is not a locally fini…

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…