# Glauberman's 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

## 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

For a complete list of replacement theorems, refer: