Jonah-Konvisser congruence condition on number of elementary abelian subgroups of small prime power order for odd prime

From Groupprops
Jump to: navigation, search
This article defines a replacement theorem
View a complete list of replacement theorems| View a complete list of failures of replacement
This article is about a congruence condition.
View other congruence conditions
This article states and (possibly) proves a fact that is true for odd-order p-groups: groups of prime power order where the underlying prime is odd. The statement is false, in general, for groups whose order is a power of two.
View other such facts for p-groups|View other such facts for finite groups

Statement

Statement in terms of a universal congruence condition

Let p be an odd prime. Suppose 0 \le k \le 5.

Let \mathcal{S} be the one-element set comprising an elementary abelian subgroup of order p^k. Then, \mathcal{S} is a Collection of groups satisfying a universal congruence condition (?) for the prime p. In particular, \mathcal{S} is a Collection of groups satisfying a strong normal replacement condition (?) for p and hence also a Collection of groups satisfying a weak normal replacement condition (?) for p.

Hands-on statement

Suppose p is an odd prime number and 0 \le k \le 5. Suppose G is a finite p-group having an elementary abelian subgroup of order p^k.

The statement has the following equivalent forms:

  1. The number of elementary abelian subgroups of G of order p^k is congruent to 1 modulo p.
  2. The number of elementary abelian normal subgroups of G of order p^k is congruent to 1 modulo p.
  3. If G is a subgroup of a finite p-group L, then the number of elementary abelian subgroups of G of order p^k that are normal in L is congruent to 1 modulo p.

In particular, if G has an elementary abelian subgroup of order p^k, then G has an elementary abelian normal subgroup of order p^k. In fact, G has an elementary abelian p-core-automorphism-invariant subgroup of order p^k, and the number of elementary abelian p-core-automorphism-invariant subgroups of G of order p^k is also congruent to 1 modulo p.

Corollary in terms of normal rank

In particular, this shows that for p an odd prime and G a p-group:

  • If the rank of G is less than or equal to 5, the normal rank of G is equal to the rank.
  • If the normal rank is at most 4, the rank equals the normal rank.

Related facts

Similar general facts

Similar replacement theorems

For a full list of replacement theorems (including many of a completely different flavor) refer Category:Replacement theorems.

Opposite facts

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

References

Journal references