Title: 2-subnormal quadratic offenders and Oliver's p-group conjecture
Author: Justin Lynd
Affliation: The Ohio State University

Abstract: Bob Oliver conjectures that if p is an odd prime and S is a
finite p-group, then the Oliver subgroup X(S) contains the Thompson
subgroup J_e(S). A positive resolution of this conjecture would give
the existence and uniqueness of centric linking systems for fusion
systems at odd primes. Using ideas and work of Glauberman, we prove
that if p \ge 5, G is a finite p-group, and V is an elementary abelian
p-group which is an F-module for G, then there exists a quadratic
offender which is 2-subnormal (normal in its normal closure) in G. We
apply this to show that Oliver's conjecture holds provided the
quotient G = S/X(S) has class at most log_2(p ??? 2) + 1, or p \ge 5 and G
is equal to its own Baumann subgroup.