Sandwich classification theorem

‎The present note arises from the author's talk at the conference ``Ischia Group Theory ۲۰۱۴''‎. ‎For subgroups F\le N of a group G denote by L(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=L(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 L(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.

کلیدواژه ها:

subgroup structure ، sandwich classification ، Chevalley groups ، commutative rings