Rank of a p-group

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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