Given a structured object "X" of any sort, a symmetry is a mapping of the object onto itself which preserves the structure. This can occur in many ways; for example, if "X" is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise to permutation groups. If the object "X" is a set of points in the plane with its metric structure or any other metric space, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (i.e., an isometry).