# 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 for which there exists an elementary abelian normal subgroup of order
- It is the largest for which there exists an abelian normal subgroup that requires a minimum of generators.

For a finite -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`

## Related notions

## 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 odd, if there exists an elementary abelian subgroup of order , the number of elementary abelian normal subgroups of order is congruent to modulo . From this, we can conclude that:

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

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

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