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 $G{L}_n(k)$ (the general linear group, the group of invertible $n\times n$ matrices with entries in $k$) is a subgroup of $G{L}_n(K)$ (the group of invertible $n\times n$ matrices with entries in $K$). This subgroup is conjugacy-closed: in other words, if two elements of $G{l}_n(k)$ are conjugate in $G{L}_n(K)$, they are also conjugate in $G{L}_n(k)$.

Related facts

Related facts about general linear group

• General linear group of subspace is conjugacy-closed
• GL IAPS is concatenation-conjugacy-closed
• Orthogonal group is conjugacy-closed in general linear group over reals
• Unitary group is conjugacy-closed in general linear group over complex numbers
• Brauer's permutation lemma

Opposite facts about general linear group

• General linear group of subring of field need not be conjugacy-closed