18.87 (2014)

A system of equations with coefficients in a group $G$ is said to be independent if the matrix composed of the sums of exponents of the unknowns has rank equal to the number of equations.
$\qquad$ a) The Kervaire–Laudenbach Conjecture (KLC): every independent system of equations with coefficients in an arbitrary group $G$ has a solution in some over group $\overline{G}$. This is true for every locally residually finite group $G$ (M. Gerstenhaber and O. S. Rothaus).
$\qquad$ b) KLC — nilpotent version: every independent system of equations with coefficients in an arbitrary nilpotent group $G$ has a solution in some nilpotent overgroup $\overline{G}$.
$\qquad$ c) KLC — solvable version: every independent system of equations with coefficients in an arbitrary solvable group $G$ has a solution in some solvable overgroup $\overline{G}$.