Sandwich classification theorem

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The present note arises from the authorʹs talk at the conference ``Ischia Group Theory 2014ʹʹ. For subgroups $Fle N$ 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 $LLʹ$ 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 $LLʹ$. If $D$ is perfect, i.,e. coincides with its commutator subgroup, then it turns out that sandwich classification theorem for $LL$ and $LLʹ$ are equivalent. We also show how to find basic subroup $F$ of sandwiches for $LLʹ$ and review sandwich classification theorems in algebraic groups over rings.