Reversibility of affine transformations

Abstract 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{GL}(n, \mathbb D) \ltimes \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{GL}(n, \mathbb D) \ltimes \mathbb D^n $ .