Abstract. The normal form of a $C^r$-differentiable loop multiplication with respect to a distinguished parametrization is investigated in case the loop is defined on the real line and the group topologically generated by the left translations is locally compact. The normal form is applied to the classification of isomorphism classes of such loops by pairs of $C^r$-differentiable real functions satisfying a differential inequality.

AMS Subject Classification (1991): 20N05, 22A30, 70G65

Keyword(s): loops, PSL_2({\msbm R}), Lie-group and Lie-algebra methods

Received December 13, 2005, and in revised form August 28, 2006. (Registered under 17/2005.)