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.
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Statement

Suppose $p$ is an odd prime, and $P$ is a $p$-group. Let $\mathcal{A}(P)$ be the set of abelian subgroups of maximum order in $P$ and $J(P)$ be the join of abelian subgroups of maximum order: the subgroup of $P$ generated by the members of $\mathcal{A}(P)$.

Suppose $B$ is a Class two normal subgroup (?) of $P$ such that its derived subgroup is contained in the center of $J(P)$ (this center is also called the ZJ-subgroup of $P$; in symbols:

$[B,B]\le Z(J(P))$.

If $A\in \mathcal{A}(P)$ is such that $B$ does not normalize $A$, there exists $A^{*}\in \mathcal{A}(P)$ such that:

• $A\cap B$ is a proper subgroup of $A^{*}\cap B$.
• $A^{*}$ normalizes $A$.

Related facts

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:

Category:Replacement theorems

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}