Abstract Algebra -- lifting up a factor group After spending an extended period with my Professor during office hours I must admit I am mystified. He kept on talking about "lifting up" factor groups. I think this has something to do with using a factor group, say G/N, to show that there exists a larger group within G itself somehow related to G/N. It would be nice if the relationship were an isomorphism but I don't think we're guaranteed that. I've been looking over all the isomorphism and homomorphism theorems especially as they relate to factor groups but am still lost. The text never uses the phrase "lifting up" and I see nothing in the text or my notes about how to go from G/N back into G itself in any useful way. Note that there are clearly trivial non-useful ways to go from G/N back into G, I just can't find a useful way. Any idea what he's talking about?