Collection of groups satisfying a property-conditional congruence condition

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Definition

Suppose $p$ is a prime number and $\mathcal{S}$ is a collection of finite $p$-groups. Suppose $\alpha$ is a property of finite $p$-groups. We say that $\mathcal{S}$ satisfies a property-conditional congruence condition for property $\alpha$ if, for any group $P$ satisfying property $\alpha$, the number of subgroups of $P$ isomorphic to elements of $\mathcal{S}$ is either zero or congruent to $1$ modulo $p$.

When $\alpha$ is the property of being any finite $p$-group, we say that $\mathcal{S}$ is a collection of groups satisfying a universal congruence condition.

Examples

Also see the examples in collection of groups satisfying a universal congruence condition.

Collection of groups	Property of ambient group	Proof
Groups of exponent $p$, class at most $p+1$	Group of exponent $p$	abelian-to-normal replacement theorem for prime exponent
Abelian groups of order $p^k$, exponent dividing $p^d$	Abelian $p$-group	congruence condition on number of subgroups of given prime power order and bounded exponent in abelian group