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.

Related notions

 * Rank of a p-group
 * Characteristic rank of a p-group

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 $$k-1$$.