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Monochromatic generating sets in groups and other algebraic structures

The \emph{generating chromatic number} of a group $G$, $\chigen(G)$, is the maximum number of colors $k$ such that there is a monochromatic generating set for each coloring of the elements of $G$ in $k$ colors. If no such maximal $k$ exists, we set $\chigen(G)=\infty$. Equivalently, $\chigen(G)$ is the maximal number $k$ such that there is no cover of $G$ by proper subgroups ($\infty$ if there is no such maximal $k$). We provide characterizations, for arbitrary gruops, in the cases $\chigen(G)=\infty$ and $\chigen(G)=2$. For nilpotent groups (in particular, for abelian ones), all possible chromatic numbers are characterized. Examples show that the characterization for nilpotent groups do not generalize to arbitrary solvable groups. We conclude with applications to vector spaces and fields.