# Class-separating field

This term is related to: linear representation theory

View other terms related to linear representation theory | View facts related to linear representation theory

*This term associates to every group, a corresponding field property. In other words, given a field, every field either has the property with respect to that group or does not have the property with respect to that group*

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

### Symbol-free definition

A field is termed **class-separating** for a group if it satisfies the following equivalent conditions:

- Given any two conjugacy classes in the group, there exists a finite-dimensional linear representation of the group over the field such that the images of the conjugacy classes, are not conjugate in the general linear group.
- Given two elements of the group whose images are conjugate in the general linear group for every finite-dimensional linear representations, the two elements must be conjugate in the group.
- No two distinct conjugacy classes can be locally conjugate.

### Definition with symbols

A field is termed **class-separating** for a group if it satisfies the following equivalent conditions:

- Given any two conjugacy classes and , there exists a finite-dimensional linear representation where is a finite-dimensional -vector space, such that and are not in the same conjugacy class in .
- Given two elements and in such that and are conjugate in for every finite-dimensional linear representation of , we can conclude that and are conjugate elements inside .

### Definition in terms of the conjugacy class-representation duality

`Further information: conjugacy class-representation duality`

Let denote the conjugacy class in of the image of the conjugacy class of under the representation . Then, is class-separating for if and only if implies that .

## Relation with other properties

### Stronger properties

### Related properties

## Facts

For a finite group, a sufficiently large field is a field of characteristic zero or relatively prime to the order of the group, which contains all the roots of unity where is the exponent of the group.

It turns out that any sufficiently large field is character-separating, and hence also class-separating.