Characteristic rank of a p-group

Definition

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

• It is the largest $r$ for which there exists an elementary Abelian characteristic subgroup of order $p^{r}$
• It is the largest $r$ for which there exists an Abelian characteristic subgroup for which the minimum size of a generating set is $r$

Related notions

• Rank of a p-group
• Normal rank of a p-group