Normal rank of a p-group

From Groupprops
Jump to: navigation, search

Definition

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

For a finite p-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 p odd, if there exists an elementary abelian subgroup of order p^k, k \le 5, the number of elementary abelian normal subgroups of order p^k is congruent to 1 modulo p. From this, we can conclude that:

  • For p odd, if the normal rank of a p-group is less than or equal to 4, the rank of that p-group is equal to the normal rank.
  • For p odd, if the rank of a p-group is less than or equal to 5, the rank equals the normal rank.

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

  • If the rank of a p-group is equal to or greater than k, so is the normal rank. Specifically, if the rank of the p-group equals k, the normal rank also equals k.
  • If the normal rank of a p-group is less than k, so is the rank. In particular, if the normal rank is k-1, the rank is also k-1.