# Splitting criterion for conjugacy classes in the special linear group

## Statement

### For a field

Suppose $K$ is a field and $n$ is a natural number. Let $G$ the general linear group $GL(n,K)$ and $H$ be the special linear group $SL(n,K)$. Then, $H$ is a normal subgroup of $G$ and is the kernel of the determinant homomorphism.

Suppose $g$ is in $H$. Then, the conjugacy class of $g$ with respect to $G$ is a subset of $H$ that is the union of one or more conjugacy classes with respect to $H$. In other words, the $G$-conjugacy class of $g$ is a union of $H$-conjugacy classes. We can obtain a bijection:

$H$-conjugacy classes in the $G$-conjugacy class of $g$ $\leftrightarrow$ the quotient group of $K^\ast$ by the image of $C_G(g)$ under the determinant map

In particular, if the image of $C_G(g)$ under the determinant map is the whole group $K^\ast$, then the $H$-conjugacy class of $g$ coincides with the $G$-conjugacy class of $g$.

### For a commutative unital ring

The statement also works if the field is replaced by a commutative unital ring.

## Facts used

1. Splitting criterion for conjugacy class in a normal subgroup