"I'm a history graduate and have a Greek Cypriot friend"
Oh wow. A Greek Cypriot friend, no less. And the depth of your knowledge on the Cyprus conflict is that "maybe the Turks didn't think that Turkey was sufficient for them and that they needed half of little Cyprus as well" Sorry, but
Without going into detail and hijacking this thread in the process, you might like to know that Turks were living in Cyprus along with Greeks. The island gained independence from the British, and there was a kind of power sharing agreement between Greek & Turkish communities (1960). President Makarios proposed unilaterally changing the constitution ("13 amendments") which Turkish Cypriots felt would diminish their political position (1963). Civil clashes ensued, and within a year, hundreds were dead on both sides. Greek Cypriots were in control of government by then, and Turkish Cypriots were living in armed enclaves, supported by Turkey.
All of which may have resolved itself, if not for one development: Military junta took power in Greece (1964) and backed a coup d'etat in Cyprus, aiming "Enosis" (unification of Cyprus with mainland Greece). Does anyone want to take a guess at what would then happen to the Turkish population, tens of thousands of whom were already displaced their homes looted, and living in shelters under armed guard?
At this point, Turkey invaded the north of the island, giving as justification the Treaty of Guarantee (1960) which bans Cyprus from any political union with any other state and requires signatories (UK, Greece, Turkey) to guarantee its independence, with force and on their own if no agreement can be reached.
Didn't your Greek Cypriot friend tell you any of this?
Personally, I think the aftermath could and should have been managed much better, and there was little reason for continued separation of the island, especially since the coup d'etat failed following Turkey's intervention and Makarios returned.
In any case, I hope you now see that Turkey's army didn't just parachute into Cyprus because they thought Turkey wasn't sufficient for them.
(End of thread hijack)