General affine group is conjugacy-closed in self-diffeomorphism group

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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 GA(n,\R) denote the general affine group: the group ofall affine transformations of \R^n, i.e. all transformations of the form:

x \mapsto Ax + b

where A is a n \times n invertible matrix over \R and b is a vector.

Let Diff(\R^n) denote the self-diffeomorphism group of \R^n. Then, any two elements of GA(n,\R), which are conjugate in Diff(\R^n), are conjugate in GA(n,\R).

In fact the above result is true even if we place Diff(\R^n) by the group of self-maps which are C^1 and have a C^1 inverse.

Related facts

It can also be shown that the general affine group is not conjugacy-closed in self-homeomorphism group. Since the property of being conjugacy-closed is a transitive subgroup property, this shows that the self-diffeomorphism group of \R^n, and in fact even the group of C^1-maps of \R^n is not conjugacy-closed in the self-homeomorphism group.

Proof

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