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_{{prime|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_{{prime|p}} is the maximal normal subgroup of G of order coprime to p, the Core (group theory)#The_p-core
O_p is the maximal normal p-subgroup of G, the p-core.
O_{{{prime|p}},p}(G) is the maximal normal p-nilpotent subgroup of G, the Core (group theory)#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 {{nowrap|p-subgroup}} of G such that PO_{{prime|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 {{nowrap|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_{{{prime|p}},p}(G) whenever P is a Sylow {{nowrap|p-subgroup}} of O_{{{prime|p}},p}(G).

{{Citation | last1=Glauberman | first1=George | author1-link=George Glauberman | title=A characteristic subgroup of a p-stable group | url=http://www.cms.math.ca/cjm/v20/p1101 |mr=0230807 | year=1968 | journal=Canadian Journal of Mathematics | issn=0008-414X | volume=20 | pages=1101–1135 | doi=10.4153/cjm-1968-107-2| doi-access=free }}
{{Citation | last1=Gorenstein | first1=D. | author1-link=Daniel Gorenstein | title=Finite Groups | publisher=Chelsea | location=New York | isbn=978-0-8284-0301-6 |mr=569209 | year=1980}}
{{Citation | last1=Thompson | first1=John G. | author1-link=John G. Thompson | title=A replacement theorem for p-groups and a conjecture | doi=10.1016/0021-8693(69)90068-4 |mr=0245683 | year=1969 | journal=Journal of Algebra | issn=0021-8693 | volume=13 | issue=2 | pages=149–151| doi-access=free }}