General linear group over subfield is conjugacy-closed

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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 k be a subfield of a field K. Then GL_n(k) (the general linear group, the group of invertible n \times n matrices with entries in k) is a subgroup of GL_n(K) (the group of invertible n \times n matrices with entries in K). This subgroup is conjugacy-closed: in other words, if two elements of Gl_n(k) are conjugate in GL_n(K), they are also conjugate in GL_n(k).

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