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

Author: Bob Oliver

Abstract: 
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.