Collection of groups satisfying a property-conditional congruence condition
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Suppose is a prime number and is a collection of finite -groups. Suppose is a property of finite -groups. We say that satisfies a property-conditional congruence condition for property if, for any group satisfying property , the number of subgroups of isomorphic to elements of is either zero or congruent to modulo .
When is the property of being any finite -group, we say that is a collection of groups satisfying a universal congruence condition.
Also see the examples in collection of groups satisfying a universal congruence condition.
|Collection of groups||Property of ambient group||Proof|
|Groups of exponent , class at most||Group of exponent||abelian-to-normal replacement theorem for prime exponent|
|Abelian groups of order , exponent dividing||Abelian -group||congruence condition on number of subgroups of given prime power order and bounded exponent in abelian group|