  These notes answer the question "When can a finite Blaschke product $B$ be
written as a composition of two finite Blaschke products $B_1$ and $B_2$, that
is, $B=B_1\circ B_2$, in a non-trivial way, that is, where the order of each is
greater than 1." It is shown that a group can be computed from $B$ and its
local inverses, and that compositional factorizations correspond to normal
subgroups of this group. This manuscript was written in 1974 but not published
because it was pointed out to the author that this was primarily a
reconstruction of work of Ritt from 1922 and 1923, who reported on work on
polynomials. It is being made public now because of recent interest in this
subject by several mathematicians interested in different aspects of the
problem and interested in applying these ideas to complex analysis and operator
theory.
