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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Let denote the general affine group: the group ofall affine transformations of , i.e. all transformations of the form:
where is a invertible matrix over and is a vector.
Let denote the self-diffeomorphism group of . Then, any two elements of , which are conjugate in , are conjugate in .
In fact the above result is true even if we place by the group of self-maps which are and have a inverse.
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 , and in fact even the group of -maps of is not conjugacy-closed in the self-homeomorphism group.