Abelian-to-normal replacement fails for half of prime plus nine for prime greater than five

History
This result is part of an as yet unpublished paper by George Glauberman.

Statement
Suppose $$p$$ is a prime number greater than $$5$$ (in other words, $$p \ge 7$$). Suppose $$k \ge (p + 9)/2$$. Then, there exists a finite $$p$$-group $$P$$ having an abelian subgroup of order $$p^k$$ but no abelian normal subgroup of order $$p^k$$.

Similar facts

 * Elementary abelian-to-normal replacement fails for half of prime plus nine for prime greater than five

Opposite facts

 * Glauberman's abelian-to-normal replacement theorem for bounded exponent and half of prime plus one
 * Mann's replacement theorem for subgroups of prime exponent
 * Elementary abelian-to-normal replacement theorem for large primes
 * Elementary abelian-to-2-subnormal replacement theorem