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
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Definition

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

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 $p^3$, $p^4$, and $p^5$ have this property for any prime number $p$. 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 $p^6$ for every prime number $p$, 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
finite abelian group