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How did the author get that $L=(L \cap H)(L\cap K)$ in Lemma $5$ below.

Remark: All the groups here are finite. $H$ permutes (commutes) with $K$ means $HK=KH$ where $H$ and $K$ are subgroups of some finite group $G$.

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Thanks in advance.

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    @DerekHolt: Thank you very much.2012-11-21

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