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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!

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  • $\begingroup$ Can you perhaps prove one of the directions of the equivalence on your own, or are you stuck on both? $\endgroup$ – zibadawa timmy Dec 3 '14 at 18:00
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    $\begingroup$ 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? $\endgroup$ – Derek Holt Dec 3 '14 at 22:03

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