Jonah-Konvisser congruence condition on number of elementary abelian subgroups of small prime power order for odd prime
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This article is about a congruence condition.
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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.
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Statement in terms of a universal congruence condition
Let be an odd prime. Suppose .
Let be the one-element set comprising an elementary abelian subgroup of order . Then, is a Collection of groups satisfying a universal congruence condition (?) for the prime . In particular, is a Collection of groups satisfying a strong normal replacement condition (?) for and hence also a Collection of groups satisfying a weak normal replacement condition (?) for .
Suppose is an odd prime number and . Suppose is a finite -group having an elementary abelian subgroup of order .
The statement has the following equivalent forms:
- The number of elementary abelian subgroups of of order is congruent to modulo .
- The number of elementary abelian normal subgroups of of order is congruent to modulo .
- If is a subgroup of a finite -group , then the number of elementary abelian subgroups of of order that are normal in is congruent to modulo .
In particular, if has an elementary abelian subgroup of order , then has an elementary abelian normal subgroup of order . In fact, has an elementary abelian p-core-automorphism-invariant subgroup of order , and the number of elementary abelian -core-automorphism-invariant subgroups of of order is also congruent to modulo .
Corollary in terms of normal rank
In particular, this shows that for an odd prime and a -group:
- If the rank of is less than or equal to , the normal rank of is equal to the rank.
- If the normal rank is at most , 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.