ZJ theorem

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_p′ is the maximal normal subgroup of G of order coprime to p, the p′-core
• O_p is the maximal normal p-subgroup of G, the p-core.
• O_p′,p is the maximal normal p-nilpotent subgroup of G, the p′,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 p-subgroup of G such that PO_p′ is normal in G, and = 1, then the image of x in NG/CG is contained in a normal p-subgroup of NG/CG.
• For an odd prime p, a group G with Op ≠ 1 is said to be p-constrained if the centralizer C_G is contained in O_p′,p whenever P is a Sylow p-subgroup of O_p′,p.