Reversibility of Affine Transformations

An element $g$ in a group $G$ is called reversible if $g$ is conjugate to $g^{-1}$ in $G$. An element $g$ in $G$ is strongly reversible if $g$ is conjugate to $g^{-1}$ by an involution in $G$. The group of affine transformations of ${\mathbb{D}}^n$ may be identified with the semi-direct product $\mathrm{G}\mathrm{L}(n,\mathbb{D})⋉{\mathbb{D}}^n$, where $\mathbb{D}:=\mathbb{R},\mathbb{C}$ or $\mathbb{H}$. This paper classifies reversible and strongly reversible elements in the affine group $\mathrm{G}\mathrm{L}(n,\mathbb{D})⋉{\mathbb{D}}^n$.