ZJ theorem

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)


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