We describe the geometry of conjugation within any split subgroup H of the full isometry group G of n -dimensional Euclidean space. We prove that, for any h H, the conjugacy class h₇ of h is described geometrically by the move-set of its linearization, while the set of elements conjugating h to a given h' h₇ is described by the fix-set of the linearization of h'. Examples include all affine Coxeter groups, certain crystallographic groups, and the group G itself.
Milićević et al. (Tue,) studied this question.