Order statistics-equivalent finite groups

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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