# Finite group in which all cumulative conjugacy class size statistics values divide the order of the group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Definition

A **finite group in which all cumulative conjugacy class size statistics values divide the order of the group** is a finite group with the following property: for every natural number , the (number of elements in for which the size of the conjugacy class of that element divides ) divides the order of .

To evaluate this property for a group, it suffices to know the conjugacy class size statistics.

## Examples

In addition to all finite abelian groups, all groups of order , , and have this property for any prime number . `Further information: All cumulative conjugacy class size statistics values divide the order of the group for groups up to prime-fifth order`

However, there exist counterexamples of order for every prime number , so the property is not held by every finite nilpotent group or even by every group of prime power order. `Further information: There exist groups of prime-sixth order in which the cumulative conjugacy class size statistics values do not divide the order of the group`

The smallest counterexample is symmetric group:S3, which is a group of order 6 where the number of elements in conjugacy classes of size dividing 3 is 4.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
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finite abelian group |