# General linear group over subfield is conjugacy-closed

From Groupprops

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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## Contents

## Statement

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

## 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