Construction of 2-local finite groups of a type studied by Solomon and Benson
Ran Levi and Bob Oliver

Abstract. A p-local finite group is an algebraic structure with a classifying
space which has many of the properties of p-completed classifying spaces 
of finite groups. In this paper, we construct a family of 2-local finite 
groups, which are exotic in the following sense: they are based on certain 
fusion systems over the Sylow 2-subgroup of Spin_7(q) (q an odd prime power) 
shown by Solomon not to occur as the 2-fusion in any actual finite group. 
Thus, the resulting classifying spaces are not homotopy equivalent to the 
2-completed classifying space of any finite group. As predicted by Benson, 
these classifying spaces are also very closely related to the Dwyer-Wilkerson 
space BDI(4).