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 O_p′(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_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(G) 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(G) ≠ 1 is said to be p-stable if whenever P is a p-subgroup of G such that PO_p′(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′,p(G) whenever P is a Sylow p-subgroup of O_p′,p(G).

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