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
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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 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 |