Unitary group is conjugacy-closed in general linear group

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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 <math>GL(n,\mathbb{C}).