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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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What primes could possibly divide $|L:(L \cap H)(L \cap K)|$? None that lie in $\pi(H)$ and none that do not lie in $\pi(H)$. –  Derek Holt Nov 19 '12 at 17:42
@DerekHolt: There is no other than one. But, I do not know how to get that from analyzing $|L:(L \cap H)(L \cap K)|$. –  user28083 Nov 20 '12 at 15:28
You know that $|L:L \cap K|$ is divisible only by primes in $\pi(H)$ and you know also that $|L:L \cap H|$ is divisible only be primes that do NOT lie in $\pi(H)$. –  Derek Holt Nov 20 '12 at 19:36
@DerekHolt: I understand your last comment. Sorry, I still don't get it. –  user28083 Nov 20 '12 at 21:04
If $|L:(L \cap H)(L \cap K)|$ is not divisible by any primes, then it must equal 1. –  Derek Holt Nov 20 '12 at 22:23

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