Sandwich classification theorem

Abstract

‎The present note arises from the author's talk at the conference ``Ischia Group Theory 2014''‎. ‎For subgroups $Fle N$ 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.