# 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.

## Definition

Two finite groups $G$ and $H$ are termed order statistics-equivalent if they satisfy the following equivalent conditions:

1. $G$ and $H$ have the same order statistics.
2. There is a bijection from $G$ to $H$ that sends any element to an element of the same order.
3. There is a finite group $K$ containing a subgroup $G_1$ isomorphic to $G$ and a subgroup $H_1$ isomorphic to $H$ such that, for any conjugacy class in $K$, its intersection with $G_1$ has the same size as its intersection with $H_1$.
4. There is a finite group $K$ containing a subgroup $G_1$ isomorphic to $G$ and a subgroup $H_1$ isomorphic to $H$ such that the induced representation of $K$ from the regular representation of $G_1$ is equivalent as a linear representation to the induced representation of $K$ from the regular representation of $H_1$.

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