Splitting criterion for conjugacy classes in the special linear group

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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.

Related facts

Similar facts

Facts used

  1. Splitting criterion for conjugacy class in a normal subgroup