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

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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 $p^4$.

## 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 $p^4$.