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

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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 $p^3$.

## 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.