Collection of groups satisfying a universal congruence condition

From Groupprops
Jump to: navigation, search
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

Definition

Suppose \mathcal{S} is a finite collection of finite p-groups, groups of prime power order for the prime p. We say that \mathcal{S} satisfies a universal congruence condition if the following equivalent conditions are satisfied by \mathcal{S}:

  1. For any finite p-group P that contains a subgroup isomorphic to an element of \mathcal{S}, the number of subgroups of P isomorphic to elements of \mathcal{S} is congruent to 1 modulo p.
  2. For any finite p-group P that contains a subgroup isomorphic to an element of \mathcal{S}, the number of normal subgroups of P isomorphic to elements of \mathcal{S} is congruent to 1 modulo p.
  3. For any finite p-group Q and any normal subgroup P of Q such that P contains a subgroup isomorphic to an element of \mathcal{S}, the number of normal subgroups of Q isomorphic to elements of \mathcal{S} and contained in P is congruent to 1 modulo p.
  4. For any finite p-group P that contains a subgroup isomorphic to an element of \mathcal{S}, the number of p-core-automorphism-invariant subgroups of P isomorphic to elements of \mathcal{S} is congruent to 1 modulo P.
  5. For any finite group G containing a subgroup isomorphic to an element of \mathcal{S}, the number of subgroups of G isomorphic to an element of \mathcal{S} is congruent to 1 modulo p.

Equivalence of definitions

Further information: equivalence of definitions of universal congruence condition

Relation with other properties

Weaker properties

Examples/facts

Satisfaction

Collection Conditions on prime p Conditions on k Proof
All groups of order p^k all p all k congruence condition on number of subgroups of given prime power order
Elementary abelian group of order p^k odd prime 0 \le k \le 5 Jonah-Konvisser congruence condition on number of elementary abelian subgroups of small prime power order for odd prime
Abelian groups of order p^k odd prime 0 \le k \le 5 Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order for odd prime
Abelian groups of order p^k, exponent dividing p^d odd prime 0 \le d \le k \le 5 Congruence condition on number of abelian subgroups of small prime power order and bounded exponent for odd prime
Abelian groups of order p^3 all p k = 3 congruence condition on number of abelian subgroups of prime-cube order
Abelian groups of order p^4 all p k = 4 congruence condition on number of abelian subgroups of prime-fourth order
Abelian groups of order 8, exponent dividing 4 p = 2 k = 3 congruence condition on number of abelian subgroups of order eight and exponent dividing four
Abelian groups of order 16, exponent dividing 8 p = 2 k = 4 congruence condition on number of abelian subgroups of order sixteen and exponent dividing eight
Non-cyclic groups of order p^3 odd p k = 3 congruence condition on number of non-cyclic subgroups of prime-cube order for odd prime

Dissatisfaction

Collection Conditions on prime p Conditions on k Proof
Klein four-group p = 2 k = 2 elementary abelian-to-normal replacement fails for Klein four-group
Elementary abelian group of order 2^k p = 2 k \ge 2 Follows from above
Elementary abelian group of order p^k all p k \ge 6
Abelian groups of order p^6 all p k = 6 Congruence condition fails for abelian subgroups of prime-sixth order
Abelian groups of order p^k all p k \ge 6 Follows from above.
Elementary abelian group of order p^6 all p k = 6 Congruence condition fails for elementary abelian subgroups of prime-sixth order (example same as for abelian subgroups)
Elementary abelian group of order p^k all p k \ge 6 Follows from above.
Groups of order p^p, exponent p all p k = p Congruence condition fails for subgroups of order p^p and exponent p

Threshold values

This lists threshold values of k: the largest value of k for which the collection of p-groups of order p^k satisfying the stated condition satisfies a universal congruence condition. The nature of all these is such that the universal congruence condition is satisfied for all smaller k but for no larger k. We use between a and b to mean that the value is at least a and at most b.

Collection of groups p = 2 p = 3 p = 5 p = 7 p \ge 11
Abelian groups of order p^k between 4 and 5 5 5 5 5
Abelian groups of order p^k, exponent dividing p^d, 2 \le d \le k between 3 and 5 5 5 5 5
Elementary abelian group of order p^k 1 5 5 5 5
Groups of exponent p, order p^k 1 2 between 2 and 4 between 2 and 6 between 2 and p - 1