Class-separating field

Definition

Symbol-free definition

A field is termed class-separating for a group if given any two conjugacy classes in the group, there exists a linear representation of the group over the field such that the images of the conjugacy classes, are not conjugate in the general linear group.

Another way of saying this is that no two distinct conjugacy classes can be locally conjugate.

Definition with symbols

A field $k$ is termed class-separating for a group $G$ if given any two conjugacy classes $c_1$ and $c_2$, there exists a representation $\rho :G\to GL(V)$ where $V$ is a $k$-vector space, such that $\rho (c_1)$ and $\rho (c_2)$ are not in the same conjugacy class in $GL(V)$.

Definition in terms of the L-notation

Further information: 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 $c_1=c_2$.

Relation with other properties

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.