Greek arkhos means chief, leader or ruler and if we add the prefix an- (without) to it we get anarchy, which literally means without a ruler and is often used to refer to a state of disorder and chaos (esp social or political, such as can be expected when there is no ruler). An anarchist, one the other hand, is a person who believes that there should be no government or governance and an arch-enemy means the chief or principal enemy.
The Aegean Sea (near Greece) was earlier called archipelagos (chief-sea, from Greek pelagos: sea). But as the Aegean is dotted with islands and island chains, archipelago began to be used to refer to a large group of islands.
Greek arkheia which referred to government office or records became archive in English which we now use as a noun to refer to a data bank or repository (as in, the film archive or the archives of the local government) and as a verb to mean to store (the document was archived in 1962).
Tyrant and tyranny (repressive and cruel rule, as in, the tyranny of greed), on the other hand, are related to Greek tyrannos (absolute ruler).
High Frequency Words for GRE, GMAT, SAT, CAT & Other Admission Tests*:
Anarchy**, Anarchist, Arch-enemy