The primary purpose of this work was to study decompositions of finite Blaschke products. We found that if a finite Blaschke productBcould be written asB∘h=Bwherehis a nontrivial holomorphic function from the unit disk into the unit disk, thenBcould be decomposed into a composition of two finite Blaschke products of lower order. It turns out thathmust be an elliptic Möbius transformation and the decomposition ofBdepends on the cycle decomposition ofhas a permutation. This is the main theorem of the paper. The first proof is classical while the second proof uses hypergroups. The existence of a nontrivial subhypergroup of the hypergroup ofBsay thatBcan be decomposed. The holomorphic functionhis used to establish this subhypergroup. We follow the development of hypergroups by Kenneth B. Stephenson [6] and use analytic continuations to present this theory.