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I have already gone thru this link Show that a group of order $108$ has a normal subgroup of order $9$ or $27$.

But i did not understand how $H \cap K$ is normal in H and K My understanding to show : g$(H \cap K) g^{-1} \in H$ it means how can we show g$K g^{-1} \in H$ Pls clarify

Magneto
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    As indicated in that answer, it is because the index of the subgroup is the smallest prime dividing the order of the group. It is a general result that such subgroups are normal. – Tobias Kildetoft Oct 11 '17 at 08:13
  • This is known as the 'Strong Calley Theorem'. See this question, https://math.stackexchange.com/questions/164244/normal-subgroup-of-prime-index . – Bysshed Oct 11 '17 at 13:44

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