# Splitting criterion for conjugacy classes in the special linear group

## Contents

## Statement

### For a field

Suppose is a field and is a natural number. Let the general linear group and be the special linear group . Then, is a normal subgroup of and is the kernel of the determinant homomorphism.

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

-conjugacy classes in the -conjugacy class of the quotient group of by the image of under the determinant map

In particular, if the image of under the determinant map is the whole group , then the -conjugacy class of coincides with the -conjugacy class of .

### For a commutative unital ring

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