Glauberman's replacement theorem
From Groupprops
This article defines a replacement theorem
View a complete list of replacement theorems| View a complete list of failures of replacement
This article states and (possibly) proves a fact that is true for odd-order p-groups: groups of prime power order where the underlying prime is odd. The statement is false, in general, for groups whose order is a power of two.
View other such facts for p-groups|View other such facts for finite groups
Contents
Statement
Suppose is an odd prime, and is a -group. Let be the set of abelian subgroups of maximum order in and be the join of abelian subgroups of maximum order: the subgroup of generated by the members of .
Suppose is a class two normal subgroup of such that its derived subgroup is contained in the center of (this center is also called the ZJ-subgroup of ) in symbols:
.
If is such that does not normalize , there exists such that:
- is a proper subgroup of .
- normalizes .
Related facts
Breakdown at the prime two
Other replacement theorems
- Thompson's replacement theorem for abelian subgroups
- Thompson's replacement theorem for elementary abelian subgroups
For a complete list of replacement theorems, refer:
Applications
- Any class two normal subgroup whose derived subgroup is in the ZJ-subgroup normalizes an abelian subgroup of maximum order
- Glauberman's theorem on intersection with the ZJ-subgroup
- p-constrained and p-stable implies Glauberman type for odd p
- Glauberman-Thompson normal p-complement theorem
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 274, Theorem 2.7, Section 8.2 (Glauberman's theorem), ^{More info}
Journal references
- A characteristic subgroup of a p-stable group by George Isaac Glauberman, , Volume 20, Page 1101 - 1135(Year 1968): ^{}^{More info}