# Order statistics-equivalent finite groups

From Groupprops

This article defines an equivalence relation over the collection of groups. View a complete list of equivalence relations on groups.

## Contents

## Definition

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.

## Facts

### 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

### Stronger 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 |