Any class two normal subgroup whose derived subgroup is in the ZJ-subgroup normalizes an abelian subgroup of maximum order
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, is a finite -group, and is a class two normal subgroup of such that its commutator subgroup is contained in the ZJ-subgroup . Then, there exists an abelian subgroup of maximum order of such that normalizes .
Related facts
- Glauberman's replacement theorem
- Thompson's replacement theorem
- Any abelian normal subgroup normalizes an abelian subgroup of maximum order: A corollary to Thompson's replacement theorem in much the same way as this result is a corollary to Glauberman's replacement theorem.
For more replacement theorems, refer Category:Replacement theorems.
Applications
Facts used
Proof
Given: An odd prime , a finite -group , a class two normal subgroup of such that .
To prove: There exists an abelian subgroup of maximum order in such that normalizes .
Proof: Let be the set of abelian subgroups of maximum order in . Let be a member of such that has maximum order.
If normalizes , we are done. Otherwise, fact (1) guarantees that there exists such that is a proper subgroup of . This contradicts the choice of as the subgroup for which has maximal order, so must normalize .
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 278, Theorem 2.9, Chapter 8 (p-constrained and p-stable groups), Section 2 (Glauberman's theorem), ^{More info}