Let $G$ be a finitely generated group with generators $X=\{g_1,g_2,\dots,g_m\}$ and $n$ be a positive integer such that the $n$-th power of every element in $X$ is the identity.
Is it true that $\mathrm{exp}(G) | n$? Clearly for abelian groups it is true. I think that it can develop to other specific groups.