I recently discovered a fun blog "A Collection of Unmitigated Pedantry", and it's been very enjoyable reading, but one of the posts is so wrong I need to write a rebuttal.
The author in Why Are There No Empires in Age of Empires? comes up with the following thesis:
An empire is a state where the core ruling population exercises control and extracts resources from a periphery which is composed of people other than the core group (linguistically/culturally/ethnically/religiously distinct). So an empire is a state where one set of people (the core) extract resources (typically by force) from another set of people (the periphery).
And so...
And that's how we know that "United States" is really a Puerto Rican Empire, as Puerto Rico is sure sucking up vastly greater flow of resources than it's ever getting back. The ruling Puerto Rican elite is literally not even paying income tax like those exploited mainlanders!
The so called "Russia" is really Great Chechen Empire, they sneakily conquered it by losing the war.
The so called "China" is really Greater Tibetan Empire.
The "United Kingdom" is really Scottish Empire, just look at all the money they're extracting from England!
And the earlier so called "British" Empire was really ran by the Americans - the exploited Brits were paying 10x as much in tax per capita than the ruling Americans.
And during Scramble for Africa, Africans managed to colonize Europe, as sure as hell average European was paying for all that elite map painting, and didn't benefit in any way whatsoever.
And so on.
This is all nonsense
If you look at any actual empires, especially recently, the overwhelming pattern is that burden of maintaining the empire - in terms of both money and manpower - falls disproportionately on core population, and peripheral populations are mostly expected to not cause trouble, and are otherwise net economic drain.
Exceptions happen, and occasionally core areas find some place extremely exploitable, but they're just that - rare exceptions. Like in 1700s' French colonial empire, tiny Guadeloupe was somehow economically worth far more than a quarter of North American continent.
Most of the time, those peripheral taxes are far less than core population pays, and far less than costs of holding those places in the Empire.
Foreign elites
There's a distinct and far more common situation where foreigners conquer some place, displace local elite, and rule as a dynasty or nobility over far more numerous locals.
This would actually fit in the original definition, except as a general rule the conquerors move to their conquered lands, so it feels a bit silly to talk about "Norman Empire" which lost Normandy rather quicky, or "Manchu Empire", which was absolutely ruled from within China, and is generally labelled "China" in history books, or such.
Exceptions proving the rule
Perhaps if one keeps looking, it's possible to find some "empires" where core successfully and sustainably extracts resources from periphery, especially in more ancient times.
Roman Empire is actually halfway there, as it managed to exploit places like Greece and Egypt pretty damn well, but even Romans admitted that their periphery like Britain were a net drain. Eventually the ruling center of Roman Empire moved to its richest province, and the usual pattern was restored.
In any case, such exceptional cases cannot possibly be used as definition of an empire.