Rank of a p-group

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


 * It is the maximum $$r$$ for which there exists an elementary abelian subgroup of order $$p^r$$.
 * It is the maximum $$r$$ for which there exists an abelian subgroup for which the minimum size of a generating set is $$r$$

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

This is also sometimes termed the depth of the p-group.

Related notions

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