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

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This article discusses a failure of replacement, i.e., a situation where the analogue of a valid replacement theorem fails to hold under slightly modified conditions.
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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This result is part of an as yet unpublished paper by George Glauberman.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.

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