Classification of saturated fusion systems on cyclic group of prime power order

Statement
Suppose $$p$$ is a prime number and $$n$$ is a natural number. Then, the cyclic group of prime power order has fusion systems in correspondence with the divisors of $$p - 1$$. Specifically, there is a correspondence:

Fusion systems on the cyclic group of order $$p^n$$ $$\leftrightarrow$$ Set of positive divisors of $$p - 1$$

The correspondence is as follows: for any divisor $$d$$ of $$p - 1$$, the automorphism group of the cyclic group of order $$p^n$$ has a unique subgroup of order $$d$$. Take the semidirect product with this subgroup and consider the fusion system induced by this.

In particular, the number of such fusion systems is the divisor count function of $$p - 1$$. It is independent of $$n$$, i.e., it is the same number for all $$n$$.

Further, in case $$p = 2$$, the number is 1, i.e., the only fusion system is the inner fusion system.

Related facts

 * Classification of saturated fusion systems on abelian group of prime power order
 * Abelian group of prime power order implies resistant