Is $HK$ a subgroup of $G$ if and only if $HK=KH$?

[Chris L T521]

Thanks to those who participated in last week's POTW! Here's this week's problem!

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Problem: Let $H$ and $K$ be subgroups of a group $G$. Define $HK = \{hk : h\in H, k\in K\}$. Show that $HK$ is a subgroup of $G$ if and only if $HK=KH$.

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[Chris L T521]

This week's question was correctly answered by Sudharaka. You can find his solution below:

Spoiler

Let \(HK\leq G\). Take any element, \(kh\in KH\). Then,

\[kh=(h^{-1}k^{-1})^{-1}\]\(h^{-1}k^{-1}\in HK\) and since \(HK\) is a group \(kh=(h^{-1}k^{-1})^{-1}\in HK\). \[\therefore KH\subseteq HK~~~~~~~~(1)\]Take any element, \(hk\in HK\). Since \(HK\) is a group there exist, \((hk)^{-1}\in HK\) such that,\[(hk)(hk)^{-1}=1\]\[\Rightarrow (hk)^{-1}=k^{-1} h^{-1}\in KH\]That is each element of \(HK\) has its' inverse in \(KH\). But since \(KH\subseteq HK\) each element of \(KH\) is also an element in \(HK\). Therefore it follows that each element of \(KH\) has its inverse in \(KH\). Now take any element \(hk\in HK\).\[hk=(k^{-1}h^{-1})^{-1}\in KH\]\[\therefore HK\subseteq KH~~~~~~~~~~~~~(2)\]By (1) and (2),\[HK=KH\]Conversely let us assume that, \(HK=KH\). Take any element \(h_{1}k_{1},\,h_{2}k_{2}\in HK\). Then,\[h_{1}k_{1}(h_{2}k_{2})^{-1}=h_{1}k_{1}k_{2}^{-1}h_{2}^{-1}\]Since \(HK=KH\) there exist \(h_{3}\in H\mbox{ and }k_{3}\in K\) such that, \(k_{1}k_{2}^{-1}h_{2}^{-1}=h_{3}k_{3}\). Therefore,\[h_{1}k_{1}(h_{2}k_{2})^{-1}=h_{1}k_{1}k_{2}^{-1}h_{2}^{-1}=h_{1}h_{3}k_{3}\in HK\]That is,\[h_{1}k_{1}(h_{2}k_{2})^{-1}\in HK\,\forall\,h_{1}k_{1},\,h_{2}k_{2}\in HK\]\[\therefore HK\leq G\]Q.E.D.