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