# Normal rank of a p-group

## Definition

The normal rank of a p-group is defined in the following equivalent ways:

• It is the largest $r$ for which there exists an elementary abelian normal subgroup of order $p^r$
• It is the largest $r$ for which there exists an abelian normal subgroup that requires a minimum of $r$ generators.

For a finite $p$-group, i.e., a group of prime power order, the normal rank is finite.

### Equivalence of definitions

Further information: Equivalence of definitions of normal rank

## Facts

The normal rank is always less than or equal to the rank. Here, we discuss the situations where equality holds.

According to the Jonah-Konvisser elementary abelian-to-normal replacement theorem, for $p$ odd, if there exists an elementary abelian subgroup of order $p^k, k \le 5$, the number of elementary abelian normal subgroups of order $p^k$ is congruent to $1$ modulo $p$. From this, we can conclude that:

• For $p$ odd, if the normal rank of a $p$-group is less than or equal to $4$, the rank of that $p$-group is equal to the normal rank.
• For $p$ odd, if the rank of a $p$-group is less than or equal to $5$, the rank equals the normal rank.

More generally, if the one-element collection of the elementary abelian group of order $p^k$ is a collection of groups satisfying a weak normal replacement condition, then:

• If the rank of a $p$-group is equal to or greater than $k$, so is the normal rank. Specifically, if the rank of the $p$-group equals $k$, the normal rank also equals $k$.
• If the normal rank of a $p$-group is less than $k$, so is the rank. In particular, if the normal rank is $k-1$, the rank is also $k-1$.