Order statistics-equivalent finite groups
This article defines an equivalence relation over the collection of groups. View a complete list of equivalence relations on groups.
Two finite groups and are termed order statistics-equivalent if they satisfy the following equivalent conditions:
- and have the same order statistics.
- There is a bijection from to that sends any element to an element of the same order.
- There is a finite group containing a subgroup isomorphic to and a subgroup isomorphic to such that, for any conjugacy class in , its intersection with has the same size as its intersection with .
- There is a finite group containing a subgroup isomorphic to and a subgroup isomorphic to such that the induced representation of from the regular representation of is equivalent as a linear representation to the induced representation of from the regular representation of .
Note that any two order statistics-equivalent finite groups have the same order.
Relation with group properties
Note that the third column (first question column) is the conjunction of the fourth and fifth columns.
|Group property||Finite version||Is any group order statistics-equivalent to a group with this property isomorphic to it?||Does any group order statistics-equivalent to a group with this property also have this property?||Are any two groups with this property that are order statistics-equivalent also isomorphic?|
|Cyclic group||Finite cyclic group||Yes: Finite group having the same order statistics as a cyclic group is cyclic||Yes||Yes|
|Abelian group||Finite abelian group||No||No; for instance: Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring||Yes: Finite abelian groups with the same order statistics are isomorphic|
|Nilpotent group||Finite nilpotent group||No||Yes: Order statistics of a finite group determine whether it is nilpotent||No; for instance: Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring|
Relation with other relations
|Relation||Meaning||Proof of implication||Proof of strictness (reverse implication failure)||Intermediate notions|
|1-isomorphic finite groups (finite case)||there is a bijection between the groups that is a 1-homomorphism of groups and whose inverse is also a 1-homomorphism of groups||1-isomorphic implies order statistics-equivalent||order statistics-equivalent not implies 1-isomorphic|
|Order-cum-power statistics-equivalent finite groups|
|Root statistics-equivalent finite groups|