Commutators inside center of factor subgroups

Let $G$ be a finite group. Assume that $K \subseteq L \trianglelefteq G$ with $K \trianglelefteq G$. Then $L /K \subseteq \textbf{Z}(G / K)$ if, and only if $[G,L] \subseteq K$.

I know it is related to the three subgroup lemma, but I don't see the relation. Thanks in advance!

• Can you perhaps prove one of the directions of the equivalence on your own, or are you stuck on both? – zibadawa timmy Dec 3 '14 at 18:00
• You don't need the three subgroup lemma. When $K=1$ it becomes $L \le Z(G) \Leftrightarrow [G,L]=1$. Can you do that case? – Derek Holt Dec 3 '14 at 22:03