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

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 fact about::collection of groups satisfying a universal congruence condition for the prime $$p$$. In particular, $$\mathcal{S}$$ is a fact about::collection of groups satisfying a strong normal replacement condition for $$p$$ and hence also a fact about::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.

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

Similar replacement theorems

 * Elementary abelian-to-normal replacement theorem for prime-square order is a weaker version proved much more easily using the same techniques.
 * Jonah-Konvisser abelian-to-normal replacement theorem

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