Splitting criterion for conjugacy classes in the special linear group
For a field
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.