ZJ theorem
Encyclopedia
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 Op′(G)Z(J(S)) is a normal subgroup
of G, for any Sylow p-subgroup S.
George Glauberman
George Glauberman is a mathematician at the University of Chicago who works on finite simple groups. He proved the ZJ theorem and the Z* theorem....
's ZJ theorem states that if a finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
G is p-constrained and p-stable and has a normal p-subgroup for some odd prime p, then Op′(G)Z(J(S)) is a normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
of G, for any Sylow p-subgroup S.
Notation and definitions
- J(S) is the Thompson subgroupThompson subgroupIn mathematical finite group theory, the Thompson subgroup J of a finite p-group P refers to one of several normal subgroups. originally defined of J to be the subgroup generated by the abelian subgroups of P of maximal order...
of a p-group S: the subgroup generated by the abelian subgroupsAbelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
of maximal orderOrder (group theory)In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
. - Z(H) means the centerCenter (group theory)In abstract algebra, the center of a group G, denoted Z,The notation Z is from German Zentrum, meaning "center". is the set of elements that commute with every element of G. In set-builder notation,...
of a group H. - Op′ is the maximal normal subgroup of G of order coprime to p, the p′-core
- Op is the maximal normal p-subgroupP-groupIn mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element...
of G, the p-core. - Op′,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 Op(G) ≠ 1 is said to be p-stable if whenever P is a p-subgroup of G such that POp′(G) is normal in G, and [P,x,x] = 1, then the image of x in NG(P)/CG(P) is contained in a normal p-subgroup of NG(P)/CG(P).
- For an odd prime p, a group G with Op(G) ≠ 1 is said to be p-constrained if the centralizer CG(P) is contained in Op′,p(G) whenever P is a Sylow p-subgroup of Op′,p(G).