# Gillam's abelian-to-normal replacement theorem for metabelian groups

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This article defines a replacement theorem

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## Statement

Suppose is a prime number and is a finite p-group. Suppose is a metabelian group, i.e., its derived length is at most two. Then, the following are true:

- For any abelian subgroup of maximum order in , there exists an abelian normal subgroup of that is contained in the normal closure of in and has the same order as .
- For any elementary abelian subgroup of maximum order in , there exists a elementary abelian normal subgroup of that is contained in the normal closure of in and has the same order as .