# Difference between revisions of "Finite group in which all cumulative order statistics values divide the order of the group"

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A '''finite group in which all cumulative order statistics values divide the order of the group''' is a [[finite group]] <math>G</math>with the following property: for every natural number <math>d</math>, the number of elements <math>g</math> such that <math>g^d</math> is the identity element is a divisor of the order of <math>G</math>. | A '''finite group in which all cumulative order statistics values divide the order of the group''' is a [[finite group]] <math>G</math>with the following property: for every natural number <math>d</math>, the number of elements <math>g</math> such that <math>g^d</math> is the identity element is a divisor of the order of <math>G</math>. | ||

− | 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. | + | In other words, a [[finite group]] in which all the values in the cumulative version of the [[defining ingredient::order statistics of a finite group|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== | ==Relation with other properties== | ||

+ | |||

+ | ===Stronger properties=== | ||

{| class="sortable" border="1" | {| class="sortable" border="1" |

## Latest revision as of 02:09, 16 June 2011

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 |