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

Statement
Suppose $$P$$ is an abelian group of prime power order with underlying prime $$p$$. Then, the following gives a classification of all the saturated fusion systems on $$P$$.

As a correspondence

 * For strict classification, i.e., not up to isomorphism of fusion systems:

Saturated fusion systems on $$P$$ $$\leftrightarrow$$ Subgroups of $$\operatorname{Aut}(P)$$ of order relatively prime to $$p$$

The correspondence works as follows: for any subgroup $$A$$ of $$\operatorname{Aut}(P)$$ of order relatively prime to $$p$$, consider the group $$G = P \rtimes A$$ and take the fusion system on $$P$$ induced by $$G$$.


 * For classification up to isomorphism of fusion systems:

Isomorphism classes of saturated fusion systems on $$P$$ $$\leftrightarrow$$ Conjugacy classes of subgroups of $$\operatorname{Aut}(P)$$ of order relatively prime to $$p$$

The correspondence works as follows: for any subgroup $$A$$ of $$\operatorname{Aut}(P)$$ of order relatively prime to $$p$$, consider the group $$G = P \rtimes A$$ and take the fusion system on $$P$$ induced by $$G$$. Note that two subgroups of $$\operatorname{Aut}(P)$$ are conjugate in $$\operatorname{Aut}(P)$$ iff they give isomorphic actions on $$P$$, hence we have to consider subgroups up to conjugacy.

Note in particular that there are no exotic fusion systems on any abelian group of prime power order.

Facts used

 * 1) uses::Identity functor controls strong fusion for saturated fusion system on abelian group