Was Greece part of the Persian Empire?

Greece was never a part of the Persian Empire. However, Greece and the Persian Empire came into conflict several times during the 5th and 4th centuries BC. The most famous conflict was the Greco-Persian Wars, which began in 499 BC when the Persians invaded Greece. The Greeks were able to defeat the Persians at the Battle of Marathon in 490 BC and the Battle of Salamis in 480 BC, and the Persians eventually withdrew from Greece.
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