Collection of groups satisfying a universal non-divisibility 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 collection of finite p-groups, i.e., groups of prime power order for a prime number p. We say that \mathcal{S} satisfies a universal non-divisibility condition if the following equivalent conditions hold:

  1. For any finite p-group P containing a subgroup isomorphic to an element of \mathcal{S}, the number of subgroups of P isomorphic to elements of \mathcal{S} is not divisible by p.
  2. For any finite p-group P containing a subgroup isomorphic to an element of \mathcal{S}, the number of subgroups of P isomorphic to elements of \mathcal{S} is not divisible by p.
  3. For any finite p-group Q and any normal subgroup P of Q containing 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 not divisible by 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 is not divisible by 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 not divisible by p.

Relation with other properties

Stronger properties

Weaker properties