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

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 fact about::collection of groups satisfying a universal congruence condition. In particular, it is also a fact about::collection of groups satisfying a strong normal replacement condition and hence also a fact about::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 $$p^k$$.

Similar general facts

 * Congruence condition on number of subgroups of given prime power order
 * Congruence condition relating number of subgroups in maximal subgroups and number of subgroups in the whole group

Generalizations

 * Congruence condition on number of abelian subgroups of small prime power order and bounded exponent for odd prime

Similar congruence condition/replacement theorems
Congruence condition-cum-replacement theorem results for odd primes:


 * Jonah-Konvisser congruence condition on number of elementary abelian subgroups of small prime power order for odd prime
 * Congruence condition on number of elementary abelian subgroups of prime-square order for odd prime
 * Congruence condition on number of elementary abelian subgroups of prime-cube and prime-fourth order for odd prime
 * Jonah-Konvisser congruence condition on number of abelian subgroups of prime-square index for odd prime

Congruence conditions for all primes:


 * Congruence condition on number of subgroups of given prime power order
 * Congruence condition on number of abelian subgroups of prime-cube order
 * Congruence condition on number of abelian subgroups of prime-fourth order
 * Congruence condition on number of abelian subgroups of prime index

Pure replacement theorems:


 * Glauberman's abelian-to-normal replacement theorem for bounded exponent and half of prime plus one
 * Elementary abelian-to-normal replacement theorem for large primes (a weaker result that is superseded by the previous result).

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

Opposite facts

 * Congruence condition fails for abelian subgroups of prime-sixth order