The Fallacy of division reference article from the English Wikipedia on 24-Jul-2004
(provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Fallacy of division

For people who check facts
A fallacy of division occurs when someone reasons logically that something that is true of a thing must also be true of its constituents. For example, an airplane may fly across the ocean, but it is illogical to believe one of its jet engines could fly across the ocean.

The opposite of this fallacy is called fallacy of composition which arises when one claims the whole is true because its component is true.