A characterization of the 2-fusion system of L4(q)L4(q)

We study a saturated fusion system FF on a finite 2-group S   having a Baumann component based on a dihedral 2-group. Assuming F=O2(F)F=O2(F), O2(F)=1O2(F)=1, and the centralizer of the component is a cyclic 2-group, it is shown that FF is uniquely determined as the 2-fusion system of L4(q1)L4(q1) for some q1≡3(mod4). This should be viewed as a contribution to a program recently outlined by M. Aschbacher for the classification of simple fusion systems at the prime 2. The corresponding problem in the component-type portion of the classification of finite simple groups (the L2(q)L2(q), A7A7 standard form problem) was one of the last to be completed, and was ultimately only resolved in an inductive context with heavy artillery. Thanks primarily to requiring the component to be Baumann, our main arguments by contrast require only 2-fusion analysis and transfer. We deduce a companion result in the category of groups.