Collection of groups satisfying a universal non-divisibility condition

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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

• Collection of groups satisfying a universal congruence condition

Weaker properties

• Collection of groups satisfying a strong normal replacement condition
• Collection of groups satisfying a weak normal replacement condition