Abelian-to-normal replacement theorem for prime-cube order

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This article defines a replacement theorem
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Statement

Statement in terms of weak normal replacement condition

Let p be a prime number. Then, the collection of abelian groups of order p^3 is a Collection of groups satisfying a weak normal replacement condition (?).

Hands-on statement

Suppose p is a prime number and P is a finite p-group. If A is an abelian subgroup of P of order p^3, there is an abelian normal subgroup B of P of order p^3.

Related facts

Facts used

  1. Existence of abelian normal subgroups of small prime power order: This states that if n \ge 1 + k(k-1)/2, then any finite p-group of order p^n has an abelian normal subgroup of order p^k.

Proof

Given: A finite p-group P of order p^n, n \ge 3, containing an abelian subgroup A of order p^3.

To prove: P contains an abelian normal subgroup B of order p^3.

Proof: If A = P, then it is normal and we can set B = A. Thus, we assume that A is a proper subgroup of P.

In this case, since n \ge 4, fact (1) tells us that P has an abelian normal subgroup of order p^3, and we are done.