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

From Groupprops

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 order statistics values divide the order of the group** is a finite group with the following property: for every natural number , the number of elements such that is the identity element is a divisor of the order of .

In other words, a finite group in which all the values in the cumulative version of the order statistics divide the order of the group. Thus, to evaluate whether this property holds for a group, we simply need to know the order statistics of the group.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

finite abelian group | ||||

finite group that is 1-isomorphic to an abelian group | ||||

finite group that is order statistics-equivalent to an abelian group | ||||

finite p-group in which the number of nth roots is a power of p for all n |