ZJ theorem explained

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 O(G)Z(J(S)) is a normal subgroup of G, for any Sylow p-subgroup S.

Notation and definitions

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