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).

References

• Glauberman, George (1968), "A characteristic subgroup of a p-stable group", Canadian Journal of Mathematics, 20: 1101–1135, doi:10.4153/cjm-1968-107-2, ISSN 0008-414X, MR 0230807
• Gorenstein, D. (1980), Finite Groups, New York: Chelsea, ISBN 978-0-8284-0301-6, MR 0569209
• Thompson, John G. (1969), "A replacement theorem for p-groups and a conjecture", Journal of Algebra, 13 (2): 149–151, doi:10.1016/0021-8693(69)90068-4, ISSN 0021-8693, MR 0245683