# Collection of groups satisfying a property-conditional congruence condition

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## Contents |

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

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.

## Examples

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

Collection of groups | Property of ambient group | Proof |
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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 |