In this paper we seek to determine whether certain finite lattices are isomorphic to interval sublattices in the subgroup lattice of some finite group and show that strong constraints are imposed on the structure of a group by the existence of such an interval. In particular given a finite lattice A, define Q(A) to be the set of pairs (H, G) such that G is a finite group, H < G, and OG(H) is isomorphic to Л or its dual. Write Q(Л) for the set of pairs (H, G) such that ICI is minimal subject to (H, G) E Q(A). One can attempt to show that for suitable choices of A and (H, G) E Q* (A), the group G is almost simple: That is, G has a unique minimal normal subgroup D, and D is a nonabelian simple group.
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