Justin Lynd

Control of fixed points and existence and uniqueness of centric linking systems

A. Chermak has recently proved that to each saturated fusion system over a finite p-group, there is a unique associated centric linking system. B. Oliver extended Chermak’s proof by showing that all the higher cohomological obstruction groups relevant to unique existence of centric linking systems vanish. Both proofs indirectly assume the classification of finite simple groups. We show how to remove this assumption, thereby giving a classification-free proof of the Martino–Priddy conjecture concerning the p-completed classifying spaces of finite groups. Our main tool is a 1971 result of the first author on control of fixed points by p-local subgroups. This result is directly applicable for odd primes, and we show how a slight variation of it allows applications for

2-subnormal quadratic offenders and Oliver's p -group conjecture

p=2

Weak closure and Oliver’s p-group conjecture

p=2 in the presence of offenders.

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

We study a saturated fusion system F on a finite 2-group S having a Baumann component based on a dihedral 2-group. Assuming F is 2-perfect with no nontrivial normal 2-subgroups, and the centralizer of the component is a cyclic 2-group, it is shown that F is uniquely determined as the 2-fusion system of L_4(q_1) for some q_1 = 3 (mod 4). 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 L_2(q), A_7 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.

Weight conjectures for fusion systems

Bob Oliver conjectures that if

Fusion systems with Benson-Solomon components

p

Extensions of the Benson-Solomon fusion systems

is an odd prime and

Weights in a Benson-Solomon block

S