Abstract: Bob Oliver conjectures that if p is an odd prime and S is a finite p-group, then the classical Thompson subgroup J(S) is always contained in a certain other characteristic subgroup X(S), now known as the Oliver subgroup. Oliver's conjecture has a module theoretic reformulation due to Green, Hethelyi, and Lilienthal. A key question arising from this reformulation is the following: for p an odd prime, G a finite p-group, and V an F_p[G]-module, does the presence of "large" abelian subgroups of G acting quadratically on V force the existence of quadratic elements in the center of G? I will present recent work on this question which, in particular, settles Oliver's conjecture when G=S/X(S) has small nilpotence class relative to p. I will also outline a counterexample to a stronger statement, obtained in joint work with David Green, which gives some indication of the subtlety of the conjecture.
