If $H\unlhd G$ and $ K\unlhd G$ and $ H \cap K= \{e\}$. Prove $xy=yx \forall x \in H \forall y \in K$

Question

If $H\unlhd G$ and $K\unlhd G$ and $H \cap K= \{e\}$. Prove $xy=yx$ $\forall x \in H \forall y \in K$

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Answer 1

Hint: concider $xyx^{-1}y^{-1}.$ Show that it belongs to $H\cap K $

Answer 2

Hint. Let $x \in H$, $y \in K$. Consider $$ [x,y] := xyx^{-1}y^{-1} $$ Ask yourself whether $[x,y]\in H$, $[x,y] \in K$.

Answer 3

Let $h\in H, k\in K$. Then $hkh^{-1}k^{-1} = h\cdot(kh^{-1}k^{-1})\in HH = H$. Similarly, $hkh^{-1}k^{-1} = (hkh^{-1})\cdot k^{-1} \in KK = K$ Then $hkh^{-1}k^{-1} \in H\cap K = \{e\}$ so $hkh^{-1}k^{-1} = e$ and $hk=kh$.