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