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Republic is a form of government. I don't understand its use in the names of countries, e.g. People's Republic of China.
Consider this: Island of China would mean Island is owned by China, it's China's subject. To me, it would make sense if it is written Republican China. I hope you understand my concern.

''People's'' is something more which I found absurd, at least in my view. Does there exist any republic consist of demons and giants and called ''Demons' Republic of Noland''.
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1-- 'Of' needn't mean possessive ('owned by'). Here, of is "used to indicate specific identity or a particular item within a category: the city of Chicago; thoughts of love.

2-- 'People' here is, I believe, a bit of Marxist terminology: "used in the possessive in Communist or left-wing countries to indicate that an institution operates under the control of or for the benefit of the people, esp. under Communist leadership: people's republic; people's army." Perhaps from: "the ordinary persons, as distinguished from those who have wealth, rank, influence, etc.: a man of the people".

(All definitions via Dictionary.com)
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Comments  
Students: Are you brave enough to let our tutors analyse your pronunciation?
Thank you very much, Mr Micawber.

It (People's Republic of China) is a name- they don't have to make sense in English usage and grammar just like in all other languages. "People's Republic of China" is just a name to used for propaganda- People's to just it for all people not a just the few as the country is in reality. The Republic of China is a not a republic- it is an oligarchy.