Vol. 26, No. 1, 1968

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On generating subgroups of the Moebius group by pairs of infinitesimal transformations

Franklin Lowenthal

Vol. 26 (1968), No. 1, 141–147

The Moebius group, i.e., the set of all transformations of the form w = (az + b)(cz + d),a,b,c and d complex numbers such that ad bc0, and its connected subgroups have been extensively studied. Its one-parameter subgroups are easily determined; the subgroup generated by a pair of such one-parameter subgroups or by their infinitesimal transformations will be defined in the usual manner. It is found that all except one, to within an inner automorphism, of the connected subgroups of the Moebius group can be generated by an appropriate pair of infinitesimal transformations. Further it is shown that the necessary and sufficient condition that a pair of infinitesimal transformations generate the entire Moebius group is that there is no Hermitian form that is left invariant by both of them. Simple criteria are given to determine whether a given pair of infinitesimal generators satisfy this condition.

Mathematical Subject Classification
Primary: 22.55
Received: 14 September 1965
Revised: 1 December 1966
Published: 1 July 1968
Franklin Lowenthal