Ore proved that a finite group is cyclic if and only if its subgroup lattice is
distributive. Now, since every subgroup of a cyclic group is normal, we call a
subfactor planar algebra cyclic if all its biprojections are normal and form a
distributive lattice. The main result generalizes one side of Ore’s theorem and shows
that a cyclic subfactor is singly generated in the sense that there is a minimal
-box
projection generating the identity biprojection. We conjecture that this result holds
without assuming the biprojections to be normal, and we show that this is true for
small lattices. We finally exhibit a dual version of another theorem of Ore and a
nontrivial upper bound for the minimal number of irreducible components for a
faithful complex representation of a finite group.
