Splitting criterion for conjugacy classes in the special linear group

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.

Similar facts

 * Splitting criterion for conjugacy classes in the alternating group

Facts used

 * 1) uses::Splitting criterion for conjugacy class in a normal subgroup