Leírás
Inverse group theory considers questions of the following form: Let F be a group-theoretic construction that produces a group F(G) from any given group G. The inverse problem is: Given a group G, is there a group H such that G = F(H)? The best studied version is where F(G) is the derived group of G, in which case a solution to the inverse problem is called an integral of G.
Although some basic questions remain unanswered – for example, we do not know whether the problem “Given a finite group G, is it integrable?” is decidable – quite a lot is known, for example, about finite and profinite groups, abelian and nilpotent groups, and varieties of groups.
A beautiful theorem of Eick gives a precise answer to the inverse Frattini problem for finite groups. Apart from this, not too much is known about other inverse problems; I will summarise some of the results which are known, concentrating mostly on finite groups.