It's exactly as Crane explains.
35mm fullframe (in German it's more correctly Kleinbild which means "small picture", today it seems huge but back in Medium Format times it was a lackluster but convenient and cheap format) is typically the reference point to which most people compare the crop factor of other formats. It's only affecting field of view, focal length always stays the same.
Basically it's like this; if you want to keep the same field of view and the same depth of field, you would need the following lenses for the different sensors:
This shows that equivalent lenses to 50mm f/1 on Leica fullframe or a 80mm f/1.9 on Mamiya medium format are just not available for the smaller formats like a 19x14mm sensor compact like the G1x or even mFT (which has an actually smaller sensor than the G1x). On the other side everyone and his dog has a 50mm f/1.8 for his SLR or a 80mm f/2.8 for his 645 or 6x6 medium format.
You could emulate the effect of the smaller sensor by just shooting with the big cam and cropping the rest away. If you think of a Nikon D700, you wouldn't expect shallow DoF when shooting a 25mm f/1.4 from 3 meter away.
Light gathering and ISO capabilities are fixed though. f/1.8 is f/1.8 for that purpose, no matter what camera. Sometimes the big sensor can be a curse therefore. While you conveniently shoot your Olympus OM-D at 20mm f/1.7 and 1/100s you would already have to shoot an EOS 5D at 40mm f/3.5 and 1/25s for an artistically identical picture with the same angle of view and DoF. You wouldn't be able to handhold the EOS at that speed though.