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

<正> 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-AE1丨≠0;g is elliptic iff g～[αβcα'],丨α丨<1 and丨E-AE1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-AE1）≠rank(E-AE1,αc-1+c-1d),where α is represented by the solutions ofcertain linear algebraic equations and satisfies丨c-1α'丨=丨（E-AE1）-1(αc-1+c-1α')丨.