4 problems

Let $G$ be a finite group, and let $R(G)$ denote its solvable radical, the largest solvable normal subgroup of $G$. For $g,a,b,c\in G$, consider the subgroup generated by $g$ and i…

Solvable-radical sequence conjecture. There exists a sequence $u_n(x,y)$ such that $R(G)$ coincides with the set of elements $y\in G$ having the property that, for every $x\in G$,…

Let $L$ be a finite-dimensional Lie algebra. For a sequence $\overrightarrow{u}=\overrightarrow{u}(x,y)$, write $L(\overrightarrow{u})$ for the corresponding ideal and let the solv…

Algorithmic radical-boundedness conjecture. There exists an algorithm which, on input $w$, decides whether there is a function