Finite group that is order statistics-equivalent to an abelian 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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Contents
Definition
A finite group is termed a finite group that is order statistics-equivalent to an abelian group is a finite group that is order statistics-equivalent (i.e., has the same order statistics) to an abelian group (in particular, a finite abelian group.
Equivalently, it is a finite nilpotent group each of whose Sylow subgroups is a group of prime power order order statistics-equivalent to an abelian 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 admitting a bijective quasihomomorphism to an abelian group|FULL LIST, MORE INFO | |||
| Finite group that is 1-isomorphic to an abelian group | |FULL LIST, MORE INFO | |||
| Group of prime power order order statistics-equivalent to an abelian group |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Finite group in which all cumulative order statistics values divide the order of the group | ||||
| Finite nilpotent group |
