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

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 $0\leq k\leq 5$. 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 $0\leq k\leq 5$. 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

• 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

References

Journal references

• Counting abelian subgroups of p-groups: a projective approach by Marc Konvisser and David Jonah, Journal of Algebra, ISSN 00218693, Volume 34, Page 309 - 330(Year 1975): ^{PDF (ScienceDirect)More info}