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.
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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==
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===Stronger properties===
  
 
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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 properties
VIEW 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 Gwith the following property: for every natural number d, the number of elements g such that g^d is the identity element is a divisor of the order of G.

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