Sandwich classification theorem

The present note arises from the author's talk at the conference ``Ischia Group Theory 2014''. For subgroups FleN of a group G denote by Lat(F,N) the set of all subgroups of N, containing F. Let D be a subgroup of G. In this note we study the lattice LL=Lat(D,G) and the lattice LL ′ of subgroups of G, normalized by D. We say that LL satisfies sandwich classification theorem if LL splits into a disjoint union of sandwiches Lat(F,N G (F)) over all subgroups F such that the normal closure of D in F coincides with F. Here N G (F) denotes the normalizer of F in G. A similar notion of sandwich classification is introduced for the lattice LL ′. If D is perfect, i.,e. coincides with its commutator subgroup, then it turns out that sandwich classification theorem for LL and LL ′ are equivalent. We also show how to find basic subroup F of sandwiches for LL ′ and review sandwich classification theorems in algebraic groups over rings.