Is THIS what you are talking about, because it's as close to China "buying" Greenland as I can find and they most certainly didn't "buy" Greenland. Australia already owned a company there and invited China to invest because they couldn't get anyone else, including the US, to bite:
Shenghe is a leading producer of rare earths in China and has acquired its stake in Greenland Minerals as a future potential source of the elements which find their way into multiple technologies, especially electric motors, as well as certain military applications such as rocket guidance systems.
The Australian company invited Shenghe to invest after it struggled to generate significant support in its home base or in Europe or North America, partly because the original plan for Greenland Mineral's Kvanefjeld project was as a source of uranium.
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