# Glauberman's abelian-to-normal replacement theorem for bounded exponent and half of prime plus one

## Statement

### Hands-on statement

Suppose $p$ is a prime number and $P$ is a finite $p$-group. Suppose $A$ is an abelian subgroup of $P$ of order $p^k$, where $k \le (p + 1)/2$ and exponent dividing $p^d$. Then, there exists an abelian normal subgroup $B$ of $P$ contained in the normal closure of $A$ such that $B$ has order $p^k$ and exponent dividing $p^d$.

### Statement in terms of normal replacement condition

Suppose $p$ is a prime number and $0 \le d \le k \le (p + 1)/2$. Then, the collection of abelian groups of order $p^k$ and exponent dividing $p^d$ is a Collection of groups satisfying a strong normal replacement condition (?).