Simple fusion systems over p-groups with abelian subgroup of index p: I

For an odd prime p, we look at simple fusion systems over a finite nonabelian p-group S which has an abelian subgroup A of index p. When S has more than one such subgroup, we reduce this to a case already studied by Ruiz and Viruel. When A is the unique abelian subgroup of index p in S and is not essential (equivalently, is not radical) in the fusion system, we give a complete list of all possibilities which can occur. This includes several families of exotic fusion systems, including some which have proper strongly closed subgroups.