ZJ theorem

In

• 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
  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
  centralizer
  C_G(P) is contained in O_p′,p(G) whenever P is a Sylow p-subgroup of O_p′,p(G).

References

• MR 0230807
• MR 0569209
• MR 0245683

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