  This paper introduces the cyclic subfactors, generalizing the cyclic groups
as the subfactors generalize the groups, and generalizing the natural numbers
as the maximal subfactors generalize the prime numbers. On one hand, a theorem
of O. Ore states that a finite group is cyclic if and only if its subgroups
lattice is distributive, and on the other hand, every subgroup of a cyclic
group is normal. Then, a subfactor planar algebra is called cyclic if all the
biprojections are normal and form a distributive lattice. The main result shows
in what sense a cyclic subfactor is singly generated, by generalizing one side
of Ore's theorem as follows: if a subfactor planar algebra is cyclic then it is
weakly cyclic (or w-cyclic), i.e. there is a minimal 2-box projection
generating the identity biprojection. Some extensions of this result are
discussed, and some applications of it are given in subfactors, quantum groups
and finite group theories, for example, a non-trivial upper bound for the
minimal number of irreducible complex representations generating the left
regular representation.
