# General linear group over subfield is conjugacy-closed

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 $GL_n(k)$.