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

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