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

This article defines a replacement theorem
View a complete list of replacement theorems| View a complete list of failures of replacement

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

• Congruence condition on number of abelian subgroups of prime-cube order
• Congruence condition on number of abelian subgroups of prime-fourth order
• Abelian-to-normal replacement theorem for prime-cube order
• Group of prime-sixth or higher order contains abelian normal subgroup of prime-fourth order for prime equal to two
• Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order for odd prime

Facts used

1. Existence of abelian normal subgroups of small prime power order: This states that if $n\geq 1+k(k-1)/2$, 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, $n\geq 4$. 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. $n\geq 7$: In this case, fact (1) tells us that there is an abelian normal subgroup of order $p^{4}$.