# Elementary Properties of Groups

Let $G$ be a group and $g,h\in G$ two elements. I need to prove that the equation $xg=h$ has a unique solution.

Applying $g^{-1}$ to both sides of the equation we get : $(xg)g^{-1}=hg^{-1}$

If I consider the Right Hand Side of the equation:

$$(xg)g^{-1}=x(gg^{-1})=x\cdot e=x$$

Now $x=hg^{-1}$ which makes me conclude that $xg=h$ has a unique solution. Can anyone correct me please!!!

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