title: ZJ theorem
text: # 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 $O_{p′,p}(G).$
  