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

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.