Glauberman's replacement theorem
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
If is such that does not normalize , there exists such that:
- is a proper subgroup of .
- normalizes .
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:
- 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
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 274, Theorem 2.7, Section 8.2 (Glauberman's theorem), More info