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Let $G$ be a finite group and let $H$ and $K$ be its proper subgroups and one is not contained in other. Suppose that $HK$ is a subgroup. Is it true, one of the $H$ or $K$ must be normal?

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closed as off-topic by Derek Holt, user91500, Claude Leibovici, Antonios-Alexandros Robotis, JonMark Perry Aug 7 '17 at 11:04

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    $\begingroup$ so it only has two proper subgroups? $\endgroup$ – Jorge Fernández Hidalgo Aug 7 '17 at 9:10
  • $\begingroup$ My guess is $H,K$ are some proper subgroups. $\endgroup$ – Alvin Lepik Aug 7 '17 at 9:15
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First interpretation: A group $G$ has only two proper subgroups $H$ and $K$. Then without loss of generality $H=\{e\}$ and so $gKg^{-1}\neq G$ anf $gKg^{-1}\neq \{e\}$ (because $g^{-1}(gKg^{-1})g=K$). We conclude $gKg^{-1}=e$.

Second interpretation: $H$ and $K$ are subgroups of $G$ such that $HK$ is a subgroup of $G$, we want to know if $H$ or $K$ are necessarily normal. This is not the case, consider $A_5$, this group is simple. If we let $H$ be the elements of $A_5$ that fix $1$ and we let $K$ be the elements of $A_5$ that fix $2$ we have that $HK=A_5$ and clearly neither is normal.

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Fact: Let $H$ and $K$ be subgroups of $G$. $HK$ is subgroup of $G$ if and only if $HK=KH$. If $H\le N_G(K)$, then $HK=KH$. The converse is not true. There are subgroups $H$ and $K$ which cannot normalizes each other, but their internal product is still a subgroup. e.g. $G=S_4$, $H=D_4$, the dihedral group, $K=\langle (123)\rangle$.

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