Any class two normal subgroup whose derived subgroup is in the ZJ-subgroup normalizes an abelian subgroup of maximum order

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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.
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Statement

Suppose p is an odd prime, P is a finite p-group, and B is a class two normal subgroup of P such that its commutator subgroup [B,B] is contained in the ZJ-subgroup Z(J(P)). Then, there exists an abelian subgroup of maximum order A of P such that B normalizes A.

Related facts

For more replacement theorems, refer Category:Replacement theorems.

Applications

Facts used

  1. Glauberman's replacement theorem


Proof

Given: An odd prime p, a finite p-group P, a class two normal subgroup B of P such that [B,B] \le Z(J(P)).

To prove: There exists an abelian subgroup A of maximum order in P such that B normalizes A.

Proof: Let \mathcal{A}(P) be the set of abelian subgroups of maximum order in P. Let A be a member of \mathcal{A}(P) such that A \cap B has maximum order.

If B normalizes A, we are done. Otherwise, fact (1) guarantees that there exists A^* \in \mathcal{A}(P) such that A \cap B is a proper subgroup of A^* \cap B. This contradicts the choice of A as the subgroup for which A \cap B has maximal order, so B must normalize A.

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