| 释义 |
- Notation and definitions
- References
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. 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.
- Op′ is the maximal normal subgroup of G of order coprime to p, the p′-core
- Op is the maximal normal p-subgroup 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).
References- {{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}}
- {{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 | pages=149–151}}
{{Abstract-algebra-stub}} 2 : Finite groups|Theorems in group theory |