Greenland is not exactly a part of Denmark as a city or as a province. Greenland belongs to the Kingdom of Denmark the same way Scotland belongs to United Kingdom.
Greenland is an autonomous constituent country within the Danish Kingdom. Greenland is geographically situated in North American continent but geopolitically it is a island of Europe.
The country its own parliament, Since 2009, Greenlad has had greater autonomy. However, its foreign policy and security are controlled by Denmark.
The native population of Greenland migrated to Greenland from Canada around 1000 BC. There is long history of Greenland how it was ruled by inhabitants from different countries for thousands of years.
In recent times, Greenland was originally ruled by Norway after union of Denmark and Norway was dissolved in 1814.
There had been controversy between Denmark and Norway over Greenland. In 1933 both countries agreed to bring the matter Permanent Court of International Justice, which ruled in favor of Denmark. Thus Greenland became a colony of Denmark. In 1953, the colony status was abolished and Greenland got autonomy.
Today, the country is mostly ruled by itself and the queen of Denmark has no power over Greenland
What happened if Greenland leave Denmark?
Denmark can survive without Greenland but Greenland would not survive without being a part of Kingdom of Denmark.
Half of the state budget of Greenland is financed by the grant from Denmark. 90% of Greenlands GDP come from fish industry. Greenland tried to explore oil and minerals without success.
If these grants are terminated in case of independence, the economy of Greenland will collapse. IMF evaluates, Greenland as a country can go bankrupt if it is separated from Denmark.
In fact, Greenland has autonomy, Only security and foreign policy are administrated by Denmark. The population of Greenland is little more than 50 thousands. The country is not capable of running it own foreign policy. Security is also a major issue.
There is another problem. Greenland is situated in North Atlantic region, not in Europe. It can not be member of EU
Denmark is the least corrupted country in the world but Greenland is not. There exits considerable corruption in Greenland. The productivity of Greenland is very low.
I do not think Greenland really wants independence. Greenland has to members in Danish parliament. They threats Denmark time to time with their independence demand and take more benefits from Denmark.
Denmark does not need Greenland but Danish politicians need them. Two parliamentary votes from Greenland determines the fate of coalition government. Without their votes, the ruling government may lose their majority.
Greenland is an economic burden for Denmark. But at the same time losing Greenland is a serious prestige issue. For this reason, Denmark does not Greenmand to leave