Suppose that $G$ is an abelian group and that I want to prove that a certain subgroup $K$ of $G$ has limited exponent and that there exists a normal subgroup $N$ of $G$ with limited exponent. Can I suppose without loss of generality that $N=\{1\}$ and to try to prove that $K$ has limited exponent?
Remark:It is not in my hypothesis that KN/N has exponent.