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
Contents
Statement
Statement in terms of universal congruence conditions
Suppose is an odd prime number, and . Then, the set of all abelian groups of order (i.e., a set of representatives of all isomorphism classes of abelian groups of order ) 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 is an odd prime number and . Suppose is a finite -group having an abelian subgroup of order . The following equivalent statements hold:
- The number of abelian subgroups of of order is congruent to modulo .
- The number of abelian normal subgroups of of order is congruent to modulo .
- If is a subgroup of a finite -group , then the number of abelian subgroups of of order that are normal in is congruent to modulo .
In particular, if has an abelian subgroup of order , then has an abelian normal subgroup of order , and moreover, has an abelian p-core-automorphism-invariant subgroup of order .
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
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
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}