On the Representation Theory of Mbius Groups in R~n

We will solve several fundamental problems of M(?)bius groups M(R~n)which havebeen matters of interest such as the conjugate classification,the establishment of a standardform without finding the fixed points and a simple discrimination method.Let g=[abcd]be a Clifford matrix of dimension n,c≠0.We give a complete conjugateclassification and prove the following necessary and sufficient conditions:g is f.p.f.(fixed pointsfree)iff g～[αβcα’],丨α丨<1 and丨E-AE~1丨≠0;g is elliptic iff g～[αβcα’],丨α丨<1 and丨E-AE~1丨=0;g is parabolic iff g～[α 0 c α’],丨α丨=1;and g is loxodromic iff g～[αβcα’],丨α丨>1 or rank(E-AE~1)≠rank(E-AE~1,αc+cd),where α is represented by the solutions ofcertain linear algebraic equations and satisfies丨cα’丨=丨(E-AE~1)(αc+cα’)丨.