Class-separating field

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


 * 1) 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.
 * 2) 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.
 * 3) No two distinct conjugacy classes can be locally conjugate.

Definition with symbols
A field $$k$$ is termed class-separating for a group $$G$$ if it satisfies the following equivalent conditions:


 * 1) Given any two conjugacy classes $$c_1$$ and $$c_2$$, there exists a finite-dimensional linear representation $$\rho:G \to GL(V)$$ where $$V$$ is a finite-dimensional  $$k$$-vector space, such that $$\rho(c_1)$$ and $$\rho(c_2)$$ are not in the same conjugacy class in $$GL(V)$$.
 * 2) Given two elements $$g$$ and $$h$$ in $$G$$ such that $$\rho(g)$$ and $$\rho(h)$$ are conjugate in $$GL(V)$$ for every finite-dimensional linear representation $$\rho$$ of $$G$$, we can conclude that $$g$$ and $$h$$ are conjugate elements inside $$G$$.

Definition in terms of the conjugacy class-representation duality
Let $$L(c,\rho)$$ denote the conjugacy class in $$GL(V)$$ of the image of the conjugacy class $$c$$ of $$G$$ under the representation $$\rho$$. Then, $$k$$ is class-separating for $$G$$ if and only if $$L(c_1,\rho) = L(c_2,\rho)$$ implies that $$c_1 = c_2$$.

Stronger properties

 * Character-separating field

Related properties

 * Class-determining field
 * Character-determining field

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 $$m^{th}$$ roots of unity where $$m$$ is the exponent of the group.

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