Orthogonal group is conjugacy-closed in general linear group over reals

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This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup (namely, Orthogonal group over reals (?)) satisfying a particular subgroup property (namely, Conjugacy-closed subgroup (?)) in a particular group or type of group (namely, General linear group over reals (?)).
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

The Orthogonal group over reals (?) O(n,\R) over the field of real numbers, is a conjugacy-closed subgroup in the General linear group over reals (?) GL(n,\R). In other words, if two orthogonal matrices are conjugate in the general linear group, they are also conjugate in the orthogonal group.

Note that the statement is not true if we replace the orthogonal group by the Special orthogonal group over reals (?).