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.

References

• Daniel Gorenstein, Finite groups, ISBN 0-8284-0301-5
• George Glauberman, A characteristic subgroup of a p-stable group, Canadian Journal of Mathematics 20 (1968), 1101–1135.
• John G. Thompson, A replacement theorem for p-groups and a conjecture, Journal of Algebra 13 (1969), 149–151 (defines the Thompson subgroup)