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

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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
This article is about a congruence condition.
View other congruence conditions
This article defines a replacement theorem
View a complete list of replacement theorems| View a complete list of failures of replacement

Statement

Statement in terms of universal congruence conditions

Suppose p is an odd prime number, and 0 \le k \le 5. Then, the set of all abelian groups of order p^k (i.e., a set of representatives of all isomorphism classes of abelian groups of order p^k) is a Collection of groups satisfying a universal congruence condition (?). In particular, it is also a Collection of groups satisfying a strong normal replacement condition (?) and hence also a Collection of groups satisfying a weak normal replacement condition (?).

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 abelian subgroup of order p^k. The following equivalent statements hold:

  1. The number of abelian subgroups of G of order p^k is congruent to 1 modulo p.
  2. The number of 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 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 abelian subgroup of order p^k, then G has an abelian normal subgroup of order p^k, and moreover, G has an abelian p-core-automorphism-invariant subgroup of order p^k.

Related facts

Similar general facts

Generalizations

Similar congruence condition/replacement theorems

Congruence condition-cum-replacement theorem results for odd primes:

Congruence conditions for all primes:

Pure replacement theorems:

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

Opposite facts

References

Journal references