lipschitz groups and lipschitz maps

‎this contribution mainly focuses on some aspects of lipschitz groups‎, ‎i.e.‎, ‎metrizable groups with lipschitz multiplication and inversion map‎. ‎in the main result it is proved that metric groups‎, ‎with a translation-invariant metric‎, ‎may be characterized as particular group objects in the category of metric spaces and lipschitz maps‎. ‎moreover‎, ‎up to an adjustment of the metric‎, ‎any metrizable abelian group also is shown to be a lipschitz group‎. ‎finally we present a result similar to the fact that any topological nilpotent element $x$ in a banach algebra gives rise to an invertible element $1-x$‎, ‎in the setting of complete lipschitz groups‎.