# Unitary group is conjugacy-closed in general linear group

From Groupprops

This article gives the statement, and proof, of a particular subgroup in a group being conjugacy-closed: in other words, any two elements of the subgroup that are conjugate in the whole group, are also conjugate in the subgroup

View a complete list of such instances/statements

## Statement

Let denote the general linear group: the group of invertible complex matrices. Let denote the unitary group: the subgroup comprising matrices such that is the identity matrix. Then, is conjugacy-closed in : any two unitary matrices that are conjugate over , are conjugate in .

## Proof

The proof uses the following facts:

- By the spectral theorem for unitary matrices, any unitary matrix is conjugate, in the unitary group, to a diagonal unitary matrix.
- Also, clearly any two diagonal unitary matrices that are conjugate in the general linear group, are conjugate by a permutation matrix, hence they are conjugate in the unitary group.

Thus, we have established a conjugate-dense subgroup of (namely the diagonal unitary matrices) such that any two elements of the subgroup are conjugate in iff they are conjugate in . This shows that is conjugacy-closed in <math>GL(n,\mathbb{C}).