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

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This article defines a replacement theorem
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Statement

Statement in terms of normal replacement condition

Let p be a prime number. The collection of abelian groups of order p^4 is a Collection of groups satisfying a weak normal replacement condition (?).

Hands-on statement

Let p be a prime number and P be a finite p-group, i.e., a group of prime power order where the prime is p. Suppose P has an abelian subgroup A of order p^4. Then, P has an abelian normal subgroup B (hence, an Abelian normal subgroup of group of prime power order (?)) of order p^4.

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.
  2. Abelian-to-normal replacement theorem for prime-square index: This states that if a finite p-group has an abelian subgroup of index p^2, it has an abelian normal subgroup of index p^2.

Proof

Given: A finite p-group P of order p^n having a subgroup A of order p^4.

To prove: P has an abelian normal subgroup B of order p^4.

Proof: Clearly, n \ge 4. We consider four cases:

  1. n = 4: This case is tautological.
  2. n = 5: In this case, any subgroup of order p^4 is of index p, hence normal, so any abelian subgroup is an abelian normal subgroup.
  3. n = 6: In this case, the result follows from fact (2).
  4. n \ge 7: In this case, fact (1) tells us that there is an abelian normal subgroup of order p^4.