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These notes answer the question "When can a finite Blaschke product $B$ bewritten as a composition of two finite Blaschke products $B_1$ and $B_2$, thatis, $B=B_1\circ B_2$, in a non-trivial way, that is, where the order of each isgreater than 1." It is shown that a group can be computed from $B$ and itslocal inverses, and that compositional factorizations correspond to normalsubgroups of this group. This manuscript was written in 1974 but not publishedbecause it was pointed out to the author that this was primarily areconstruction of work of Ritt from 1922 and 1923, who reported on work onpolynomials. It is being made public now because of recent interest in thissubject by several mathematicians interested in different aspects of theproblem and interested in applying these ideas to complex analysis and operatortheory.

Finite Blaschke Products as Compositions of Other Finite Blaschke Products