Unitary group is conjugacy-closed in general linear group

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
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Statement

Let $GL(n,\mathbb{C})$ denote the general linear group: the group of invertible $n \times n$ complex matrices. Let $U(n,\mathbb{C})$ denote the unitary group: the subgroup comprising matrices $A$ such that $AA^*$ is the identity matrix. Then, $U(n,\mathbb{C})$ is conjugacy-closed in $GL(n,\mathbb{C})$: any two unitary matrices that are conjugate over $GL(n,\mathbb{C})$, are conjugate in $U(n,\mathbb{C})$.

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 $U(n,\mathbb{C})$ (namely the diagonal unitary matrices) such that any two elements of the subgroup are conjugate in $GL(n,\mathbb{C})$ iff they are conjugate in $U(n,\mathbb{C})$. This shows that $U(n,\mathbb{C})$ is conjugacy-closed in [itex]GL(n,\mathbb{C}).