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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

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    $\begingroup$ 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. $\endgroup$ – Tobias Kildetoft Oct 11 '17 at 8:13
  • $\begingroup$ This is known as the 'Strong Calley Theorem'. See this question, math.stackexchange.com/questions/164244/… . $\endgroup$ – Bysshed Oct 11 '17 at 13:44

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