Pedant head on.
Internal combustion engine (Otto) cycle - suck, Squeeze, bang, fart.
Gas turbine (I refuse to call it Brayton as Mr Whittle formulated it years before) cycle - Suck, squeeze, burn, blow.
Just sayin'.
Anyway, to answer the original question...
If the turbine engine didn't have a compressor, then the turbine is just a spinning disc getting in the way. It has to drive something.
I suppose in theory you could have a turbo-prop with no compressor and rely on a very efficient ram effect for compression but as no-one has done it yet I suspect it won't work, for all the very clever and frankly complex answers given on the last 3 pages.
