greetings sebsi...
i have messed with a few of these... if your core just came out of the cylinder like you said you maybe missing the last spring loaded tumbler... there are 7 total... the 1st 6 are what allow the key to rotate the core in the cylinder to lock and unlock... the last tumbler is what secures the core to the cylinder...
you have to make sure the little springs are in the core and the tumblers are freely bouncing on the springs...
shine a light down the cylinder and you will see a channel... you put your key in the core and slide the core into the cylinder with the tumblers lined up with the channel... push it all the way in... once in all the way you turn it either clockwise or counter clockwise until it stops... this seats the last tumbler behind the holder...
if you look behind another lock you will see that last tumbler sitting against the black plastic holder at the end of the cylinder...
to remove the lock you have to get a small something like a tiny allen wrench or tiny screwdriver in the back to press down that last tumbler and rotate the key and it will come right out...
you may wanna take a good one out so you can see what you got... just dont lose the tumblers or springs cause you will be be f ed...
j o