ZJ theorem

In mathematics, George Glauberman's ZJ theorem states that if a finite group G is p-constrained and p-stable and has a normal p-subgroup for some odd prime p, then OZ is a normal subgroup of G, for any Sylow p-subgroup S.

Notation and definitions

J is the Thompson subgroup of a p-group S: the subgroup generated by the abelian subgroups of maximal order.
Z means the center of a group H.
O_' is the maximal normal subgroup of G of order coprime to p, the '-core
O_p is the maximal normal p-subgroup of G, the p-core.
O_',p is the maximal normal p-nilpotent subgroup of G, the ',p-core, part of the upper p-series.
For an odd prime p, a group G with O_p ≠ 1 is said to be p-stable if whenever P is a of G such that PO is normal in G, and = 1, then the image of x in N_G/C_G is contained in a normal of N_G/C_G.
For an odd prime p, a group G with O_p ≠ 1 is said to be p-constrained if the centralizer C_G is contained in O_',p whenever P is a Sylow of O_',p.