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 − bc≠0, 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.
